Combinatorial Matrix Theory

Home , Adjacency matrix, Alternating sign matrix, Hurwitz matrix, Stieltjes matrix

Richard A. Brualdi • Ángeles Carmona P. van den Driessche • Stephen Kirkland Dragan Stevanović

Editors for this volume: Andrés M. Encinas, Universitat Politècnica de Catalunya Margarida Mitjana, Universitat Politècnica de Catalunya Richard A. Brualdi Ángeles Carmona Mathematics Department Departament de Matemàtiques University of Wisconsin Universitat Politècnica de Catalunya Madison, WI, USA Barcelona, Spain

P. van den Driessche Stephen Kirkland Department of Mathematics and Statistics Department of Mathematics University of Victoria University of Manitoba Victoria, BC, Canada Winnipeg, MB, Canada

Dragan Stevanović Mathematical Institute Serbian Academy of Sciences and Arts Belgrade, Serbia

ISSN 2297-0304 ISSN2297-0312 (electronic) Advanced Courses in Mathematics - CRM Barcelona ISBN 978-3-319-70952-9 ISBN 978-3-319-70953-6 (eBook) https://doi.org/10.1007/978-3-319-70953-6

Library of Congress Control Number: 2018935110

This book contains the notes of lectures delivered by Richard A. Brualdi, Angeles´ Carmona, Stephen Kirkland, Dragan Stevanovićand Pauline van den Driessche at the Centre de Recerca Matemàtica(CRM) in Bellaterra, Barcelona from June 29th to July 3rd, 2015. The Advanced Course on Combinatorial Matrix Theory was mainly addressed to PhD students and post-docs in related areas, and also to more established researchers who wanted to get involved in these subjects. Combinatorial matrix theory is a rich branch of matrix theory; it is both an active field of research and a widespread toolbox for many scientists. Combina- torial properties of matrices are studied on the basis of qualitative rather than quantitative information, so that the ideas developed can provide consistent information about a model even when the data is incomplete or inaccurate. The theory behind qualitative methods can also contribute to the development of effective quantitative matrix methods. The topics covered in the Advanced Course included permutation matrices, alternating sign matrices, tournaments, sign pattern matrices, minimum rank and its distribution, boundary value problems on finite networks, the group inverse for the Laplacian matrix of a graph, and bounds on the spectral radius of the Laplacian matrix. The activity consisted of five series of four lectures each. Accordingly, the material is divided into five chapters in the book. There were also two short sessions devoted to informal presentations of the current work of most of the participants. The first chapter corresponds to the lectures delivered by Richard A. Brualdi on some combinatorially defined matrix classes, specifically on three families, namely permutation, alternating sign, and tournament matrices. Permutation matrices are the matrices associated with permutations, and it is shown that the Bruhat order on this symmetric group is related to Gaussian elimination and leads to the Bruhat decomposition of a nonsingular matrix. Alternating sign matrices are generalizations of permutation matrices, and the extension of Bruhat order is a lattice, specifically its McNielle completion. A tournament matrix is the adjacency matrix corresponding to a tournament; that is, an orientation of the complete graph. The generation problem is presented and analyzed on loopy, Hankel, and skew-Hankel tournaments. The second chapter is devoted to the series of lectures given by Pauline van den Driessche on sign pattern matrices. It discusses the study of spectral properties of matrices with a given sign pattern. Several classes of sign patterns are reviewed, including sign patterns that allow all possible spectra; those allowing all possible inertias; those allowing stability; and those that may give rise to Hopf bifurcation in associated dynamical systems. Some classes of these matrices are explored in more detail using techniques from matrix theory, graph theory, and analysis. Moreover, some open problems are suggested to encourage further work. Dragan Stevanovićdelivered lectures on the spectral radius of graphs, col- lected in Chapter 3. The spectral radius of a graph is a tool to provide bounds of parameters related to the properties of a graph. More precisely, eigenvalues and eigenvectors of graph matrices have become standard mathematical tools nowa- days due to their wide applicability in network analysis and computer science, with the most prominent graph matrices being the adjacency and the Laplacian matrix. In this chapter, lower and upper limits of the spectral radius of adjacency and Laplacian matrices are addressed, with special attention to testing techniques and common properties of shapes of the boundaries. In addition, approximate formulas for the spectral radius of the adjacency matrix are also discussed. The fourth chapter contains the lectures delivered by Stephen Kirkland on the group inverse for the Laplacian matrix of a graph. In recent times, Laplacian matrices for undirected graphs have received a good deal of attention, in part because the spectral properties of the Laplacian matrix are related to a number of features of interest of the underlying graph. It turns out that a certain generalized inverse, the group inverse, of a Laplacian matrix also carries information about the graph in question. This chapter explores the group inverse of the Laplacian matrix and its relationship to graph structure. Connections with the algebraic connectivity and the resistance distance are made, and the computation of the group inverse of a Laplacian matrix is also considered from a numerical viewpoint. The last chapter, authored by Angeles´ Carmona, is devoted to the study of boundary value problems on finite networks. The starting point is the description of the basic difference operators: the derivative, gradient, divergence, curl and Laplacian, or more generally, Schrödingeroperators. The next step is to define the discrete analogue of a manifold with a boundary, which includes the concept of outer normal field and proves the Green identities. At that point, the focus is on some aspects of discrete potential theory. The discrete analog of the Green and Poisson kernels are defined and their relationship with the so-called Dirichlet- to-Neuman map established. Finally, some applications to matrix theory and to organic chemistry, such as the M-inverse problem and the Kirchhoff index computation, are considered. We would like to express our gratitude to the director, Prof. Joaquim Bruna, and staff of the Centre de Recerca Matemàticanot only for their excellent job in organizing this course but also for their kindness and support during the event. We are also in debt to Elsevier, the Societat Catalana de Matemàtiques,and the Real Sociedad MatemáticaEspañolafor their financial support. Finally, we thank all the participants for their active involvement and spe- cially to the five lecturers for their accurate preparation of these notes. We hope that their publication will contribute to increasing knowledge of combinatorial matrix theory. AndrésM. Encinas and Margarida Mitjana Contents

Foreword v

1 Some Combinatorially Deﬁned Matrix Classes 1 By Richard A. Brualdi

1.1 Permutations and Permutation Matrices ...... 1 1.1.1 Basic Properties ...... 1 1.1.2 Generation ...... 3 1.1.3 Bruhat Order ...... 3 1.1.4 Matrix Bruhat Decomposition ...... 5 1.1.5 Flags ...... 8 1.1.6 Involutions and Symmetric Integral Matrices ...... 11 1.2 Alternating Sign Matrices ...... 15 1.2.1 Basic Properties ...... 15 1.2.2 Other Views of ASMs ...... 16 1.2.3 The λ-determinant ...... 19 1.2.4 Maximal ASMs ...... 20 1.2.5 Generation ...... 21 1.2.6 MacNeille Completion and the Bruhat Order ...... 22 1.2.7 Bruhat Order Revisited ...... 25 1.2.8 Spectral Radius of ASMs ...... 31 1.3 Tournaments and Tournament Matrices ...... 32 1.3.1 The Inverse Problem ...... 33 1.3.2 Generation ...... 34 1.3.3 Loopy Tournaments and Their Generation ...... 36 1.3.4 Hankel Tournaments and Their Generation ...... 38 1.3.5 Combinatorially Skew-Hankel Tournaments and Their Generation ...... 40 Bibliography ...... 45 2 Sign Pattern Matrices 47 By P. van den Driessche

2.1 Introduction to Sign Pattern Matrices ...... 47 2.1.1 Notation and Definitions ...... 47 2.2 Potential Stability of Sign Patterns ...... 48 2.2.1 Stability Definitions ...... 48 2.2.2 Stability of a Dynamical System ...... 49 2.2.3 Characterization of Sign Stability ...... 50 2.2.4 Basic Facts for Potential Stability ...... 51 2.2.5 Known Results on Potential Stability for Small Orders . . . 52 2.2.6 Sufficient Condition for Potential Stability ...... 52 2.2.7 Construction of Higher-Order Potentially Stable Sign Patterns ...... 54 2.2.8 Number of Nonzero Entries ...... 56 2.2.9 Open Problems Related to Potential Stability ...... 57 2.3 Spectrally Arbitrary Sign Patterns ...... 57 2.3.1 Some Definitions Relating to Spectra of Sign Patterns . . . 57 2.3.2 A Family of Spectrally Arbitrary Sign Patterns ...... 59 2.3.3 Minimal Spectrally Arbitrary Patterns and Number of Nonzero Entries ...... 61 2.3.4 Reducible Spectrally Arbitrary Sign Patterns ...... 62 2.3.5 Some Results on Potentially Nilpotent Sign Patterns . . . . 63 2.3.6 Some Open Problems Concerning SAPs ...... 64 2.4 Refined Inertia of Sign Patterns ...... 64 2.4.1 Definition and Maximum Number of Refined Inertias . . . . 64 2.4.2 The Set of Refined Inertias Hn ...... 65 2.4.3 Sign Patterns of Order 3 and H3 ...... 66 2.4.4 Sign Patterns of Order 4 and H4 ...... 66 2.4.5 Sign Patterns with All Diagonal Entries Negative ...... 67 2.4.6 Detecting Periodic Solutions in Dynamical Systems . . . . . 68 2.4.7 Some Open Problems Concerning Hn ...... 72 2.5 Inertially Arbitrary Sign Patterns ...... 73 2.5.1 Definition and Relation to Other Properties ...... 73 2.5.2 Generalization of the Nilpotent-Jacobian Method ...... 74 2.5.3 Reducible IAPs ...... 76 2.5.4 A Glimpse at Zero-Nonzero Patterns ...... 76 2.5.5 A Taste of More General Patterns ...... 77 2.5.6 Some Open Problems Concerning IAPs ...... 78 Bibliography ...... 79 3 Spectral Radius of Graphs 83 By Dragan Stevanović 3.1 Graph-Theoretical Definitions ...... 83 3.2 The Adjacency Matrix and Its Spectral Properties ...... 85 3.3 The Big Gun Approach ...... 89 3.4 The Eigenvector Approach ...... 95 3.5 The Characteristic Polynomial Approach ...... 105 3.6 Walk Counting ...... 110 Bibliography ...... 127

4 The Group Inverse of the Laplacian Matrix of a Graph 131 By Stephen Kirkland 4.1 Introduction ...... 131 4.2 The Laplacian Matrix ...... 132 4.3 The Group Inverse ...... 138 4.4 L# and the Bottleneck Matrix ...... 141 4.5 L# for Weighted Trees ...... 144 4.6 Algebraic Connectivity ...... 148 4.7 Joins ...... 152 4.8 Resistance Distance ...... 154 4.9 Computational Considerations ...... 160 4.10 Closing Remarks ...... 168 Bibliography ...... 171

5 Boundary Value Problems on Finite Networks 173 By Angeles´ Carmona 5.1 Introduction ...... 173 5.2 The M-Matrix Inverse Problem ...... 174 5.3 Difference Operators on Networks ...... 176 5.3.1 Schrödinger Operators ...... 181 5.4 Glossary ...... 184 5.5 Networks with Boundaries ...... 185 5.6 Self-Adjoint Boundary Value Problems ...... 188 5.7 Monotonicity and the Minimum Principle ...... 193 5.8 Green and Poisson Kernels ...... 195 5.9 The Dirichlet-to-Robin Map ...... 199 5.10 Characterization of Symmetric M-Matrices as Resistive Inverses . 201 5.10.1 The Kirchhoff Index and Effective Resistances ...... 202 5.10.2 Characterization ...... 204 5.11 Distance-regular Graphs with the M-Property ...... 207 5.11.1 Strongly Regular Graphs ...... 209 5.11.2 Distance-regular Graphs with Diameter 3 ...... 210 Bibliography ...... 215 Chapter 1

Some Combinatorially Deﬁned Matrix Classes by Richard A. Brualdi

1.1 Permutations and Permutation Matrices

In this section we consider the symmetric group of permutations of a ﬁnite set and their partial order known as the Bruhat order. Regarding a permutation as a permutation matrix, this partial order is related to Gaussian elimination and leads to the matrix Bruhat decomposition of a nonsingular matrix, and then to a characterization of ﬂags in a vector space. We also describe a correspondence between permutations that are involutions (symmetric permutation matrices) and a certain class of nonnegative integral matrices.

1.1.1 Basic Properties

One of the most basic concepts in mathematics is that of a permutation of a finite set. Let σ be a permutation of {1, 2, . . . , n}, where we write σ = (i1, i2, . . . , in) to denote that σ(k) = ik for k = 1, 2, . . . , n. Here, i1, i2, . . . , in are distinct and {i1, i2, . . . , in} = {1, 2, . . . , n}. We denote the set of all permutations of {1, 2, . . . , n} by Sn. Each permutation σ of {1, 2, . . . , n} can be identified with an n × n permutation matrix P = Pσ = [pij], where p1i1 = p2i2 = ··· = pnin = 1 and pij = 0 otherwise. 2 Chapter 1. Some Combinatorially Defined Matrix Classes

Example 1.1.1. With n = 5 and σ = (3, 5, 4, 2, 1) we have

 0 0 1 0 0   1   0 0 0 0 1   1      P =  0 0 0 1 0  or, as we often write, P =  1  ,      0 1 0 0 0   1  1 0 0 0 0 1 blocking the rows and columns and suppressing the zeros.

Let σ = (i1, i2, . . . , in) be a permutation. An ascent of σ is a pair k, k+1 with ik < ik+1. We also say that an ascent occurs at position k. Similarly, a descent of σ is a pair k, k +1 with ik > ik+1, and we also say that a descent occurs at position k. The permutation σ in Example 1.1.1 has an ascent occuring at position 1 and descents occuring at positions 2, 3, and 4. In general the number of ascents plus the number of descents in a permutation in Sn equals n − 1. An inversion in a permutation σ = (i1, i2, . . . , in) is less restrictive than a descent; it is a pair k, l of positions with k < l such that ik > il. Thus an inversion corresponds to a pair of integers in σ which are out of the natural order 1, 2, . . . , n. In terms of the permutation matrix P , an inversion corresponds to a 2 × 2 submatrix of P ,

0 1 P [k, l | i , i ] = = Q , l k 1 0 2 in rows k and l, and columns il and ik. The transposition that interchanges ik and il in σ, and thus replaces the 2×2 submatrix P [k, l | il, ik] in the permutation matrix P with the 2 × 2 identity matrix I2, is a new permutation with fewer inversions, perhaps a lot fewer. But if each of the integers il+1, il+2, . . . , ik−1 is either smaller than il or larger than ik, then the transposition decreases the number of inversions by exactly one. In terms of the permutation matrix P , if the (l −k −1)×(l −k −1) submatrix of P determined by rows k+1, . . . , l−1 and columns il+1, il+2, . . . , ik−1 is a zero matrix, then the replacement of P [k, l | ik, il] = Q2 with I2 decreases the number of inversions by exactly one. Example 1.1.2. The permutation σ in Example 1.1.1 has eight inversions. The identity matrix In (corresponding to the permutation σ = (1, 2, . . . , n)) has no inversions. The anti-identity matrix Qn (corresponding to the permutation σ = n 2 (n, n−1,..., 1)) has the maximum number 2 of inversions and satisﬁes Qn = In. For example,  1   1    Q5 =  1     1  1 1.1. Permutations and Permutation Matrices 3

5 has 2 = 10 inversions. The transposition given by

 1   1    Q5 →  1  ,    1  1 decreases the number of inversions from ten to three. Now consider the permutation τ ∈ S9 equal to (5, 8, 2, 1, 9, 3, 4, 7, 6) and its inversion 8 > 3. Since each of the integers in between 8 and 3 in τ is either greater than 8 or less than 3, the transposition (5, 8, 2, 1, 9, 3, 4, 7, 6) → (5, 3, 2, 1, 9, 8, 4, 7, 6) reduces the number of inversions by exactly one, from 16 to 15.

1.1.2 Generation

Let σ = (i1, i2, . . . , in) be any permutation of {1, 2, . . . , n}. In general, a transposition applied to σ interchanges any two integers in σ:

(i1, . . . , ik, . . . il, . . . , in) −→ (i1, . . . , il, . . . ik, . . . , in).

Obviously, starting from the permutation σ, by a sequence of transpositions we can obtain the identity permutation (1, 2, . . . , n). Reversing these transpositions, starting from the identity permutation, we can obtain any permutation. We thus have the following result that all the permutations in Sn can be generated from the identity permutation by a sequence of transpositions.

Theorem 1.1.3. Each permutation in Sn can be obtained from the identity permutation by a sequence of transpositions. More generally, given any two permutations σ and τ in Sn, we can obtain τ by a sequence of transpositions starting with σ.

1.1.3 Bruhat Order

1 In this section we deﬁne a partial order on Sn called the Bruhat order and discuss some of its important properties. Let σ and τ be permutations of {1, 2. . . . , n}. Then σ is less than or equal to τ in the Bruhat order, written σ B τ, provided σ can be obtained from τ by a sequence of inversion-reducing transformations. The Bruhat order is clearly a partial order on Sn, and it is graded by the number of inversions with grades from n 0 to 2 . It follows that the identity permutation ιn (identity matrix In) is the unique minimal permutation in the Bruhat order (having no inversions) and the anti-identity ζn (anti-identity matrix Qn) is the unique maximal permutation in n the Bruhat order (having 2 inversions).

1Fran¸coisBruhat (8 April 1929 – 17 July 2007). 4 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

 0 0 1  L3 =  0 1 0  : (3, 2, 1) 1 0 0

 0 1 0   0 0 1  (2, 3, 1) :  0 0 1   1 0 0  : (3, 1, 2) 1 0 0 0 1 0

 0 1 0   1 0 0  (2, 1, 3) :  1 0 0   0 0 1  : (1, 3, 2) 0 0 1 0 1 0

 1 0 0  I3 =  0 1 0  : (1, 2, 3) 0 0 1

Figure 1.1: Hasse diagram of (S3, B).

Example 1.1.4. Fig. 1.1 shows the Hasse diagram of the Bruhat order on S3 using both permutation and permutation matrix labels. Note that the Bruhat order (S3, B) is not a lattice, since the permutations (2, 3, 1) and (3, 1, 2) do not have a meet (a unique greatest lower bound, that is, an element which is a lower bound and is greater than every other lower bound); the permutations (2, 1, 3) and (1, 3, 2) are both greatest lower bounds of (2, 3, 1) and (3, 1, 2), but they are incomparable in the Bruhat order.

Using the deﬁnition of the Bruhat order, it is not easy to determine in general whether or not two permutations in Sn are related in the Bruhat order. But there is another characterization of the Bruhat order involving the comparison of only 2 (n − 1) integers. Let A = [aij] be any real m × n matrix and let σij(A) = Pi Pj k=1 l=1 akl, 1 ≤ i ≤ m, 1 ≤ j ≤ n, be the sum of the entries in the leading i × j submatrix of A. Let Σ(A) = [σij(A)], an m × n matrix. Note that if P is an n × n permutation matrix, then the last column of Σ(P ) as well as the last row of Σ(P ) consists of the integers 1, 2, . . . , n in that order. The following gives an easily checkable characterization of the Bruhat order; see, e.g., Bj˝orner–Brenti [2] and Magyar [25].

Theorem 1.1.5. Let σ and τ be permutations of {1, 2, . . . , n} with permutation matrices P and Q, respectively. Then, σ B τ if and only if Σ(Q) ≥ Σ(P ) entrywise.

Example 1.1.6. Consider the permutations ρ = (2, 5, 1, 3, 6, 4), τ = (4, 5, 6, 1, 2, 3), 1.1. Permutations and Permutation Matrices 5 and π = (1, 5, 3, 6, 2, 4) with corresponding permutation matrices

 1   1   1   1       1   1  Pρ =   ,Pτ =   ,  1   1       1   1  1 1 and  1   1     1  Pπ =   .  1     1  1 Then,

 0 1 1 1 1 1   0 0 0 1 1 1   0 1 1 1 2 2   0 0 0 1 2 2       1 2 2 2 3 3   0 0 0 1 2 3  Σ(Pρ) =   , Σ(Pτ ) =   ,  1 2 3 3 4 4   1 1 1 2 3 4       1 2 3 3 4 5   1 2 2 3 4 5  1 2 3 4 5 6 1 2 3 4 5 6 and  1 1 1 1 1 1   1 1 1 1 2 2     1 1 2 2 3 3  Σ(Pπ) =   .  1 1 2 2 3 4     1 2 3 3 4 5  1 2 3 4 5 6

We see that Σ(Pρ) ≥ Σ(Pτ ) and Σ(Pπ) ≥ Σ(Pτ ), and hence that ρ B τ and π B τ. But σ3,2(Pρ) = 2 > 1 = σ3,2(Pπ) and σ1,1(Pπ) = 1 > 0 = σ1,1(Pρ), and hence ρ and π are incomparable in the Bruhat order.

1.1.4 Matrix Bruhat Decomposition Let A be an n×n nonsingular complex matrix. Since A is nonsingular, in applying Gaussian elimination to reduce A to row echelon form, we pivot on n positions corresponding to a permutation set of places. Thus there is an n × n permutation matrix R such that, in applying Gaussian elimination to RA, we pivot sequentially on the main diagonal positions of RA and this results in an n×n upper triangular matrix U. The sequence of pivot operations on RA is equivalent to multiplication 6 Chapter 1. Some Combinatorially Deﬁned Matrix Classes of RA on the left by a lower triangular matrix K. Thus KRA = U, and hence A = P LU where L is the inverse of K (so, L is also lower triangular) and P is the permutation matrix R−1. We state this result as a theorem; see, e.g., Brualdi [4] and Tyrtyshnikov [27]. Theorem 1.1.7. If A is an n × n nonsingular complex matrix, then there exist a permutation matrix P , a lower triangular matrix L, and an upper triangular matrix U such that A = P LU. Since the choice of pivots in Gaussian elimination is not uniquely determined in general, the matrix P and hence the matrices L and U are not unique. Example 1.1.8. Let A be the nonsingular matrix

1 2 A = . 1 1

Then, 0 1 1 0 1 1 A = . 1 0 1 1 0 1 We also have 1 0 1 0 1 2 A = . 0 1 1 1 0 −1 There is another decomposition of a nonsingular complex matrix A with the same ingredients as PLU but a diﬀerent order of them, namely A = LP U (a lower triangular matrix followed by a permutation matrix followed by an upper triangular matrix) or, equivalently, A = LP L (a lower triangular matrix followed by a permutation matrix followed by another not necessarily the same lower triangular ma- 2 trix). Since an upper triangular matrix U equals Qn times a lower triangular matrix, this follows from AQn = LP U = LP InU = LP (QnQn)U = L(PQn)(QnU), 0 0 and hence A = L(PQn)(QnUQn) = LP L . Example 1.1.9. Consider the nonsingular matrix

 0 1 1  A =  1 2 2  . 1 2 3

Then,

 1 0 0   0 1 0   1 1 1   1 0 0   0 1 0   1 0 0  A =  1 1 0   1 0 0   0 1 1  =  1 1 0   0 0 1   1 1 0 . 1 1 1 0 0 1 0 0 1 1 1 1 1 0 0 1 1 1 2We use notations like A = LP L in a symbolic sense to denote a product of a lower triangular matrix, a permutation matrix, and another lower triangular matrix. The two lower triangular matrices need not be identical. 1.1. Permutations and Permutation Matrices 7

Either of the decompositions A = LP U and A = LP L is called the matrix Bruhat decomposition for reasons now to be explained. Theorem 1.1.10. If A is an n × n nonsingular complex matrix, then there exists a unique permutation matrix P , a lower triangular matrix L, and an upper triangular matrix U such that A = LP U. Similarly, there exist a unique permutation matrix P and two lower triangular matrices L such that A = LP L. Proof. We show that, using elementary row operations (multiplying on the left by a lower triangular matrix) and elementary column operations (multiplying on the right by an upper triangular matrix), A can be reduced to a permutation matrix, and this gives A = LP U. Here is the algorithm that does this: (i) Consider the first nonzero in row 1, the pivot element. Postmultiplication by an upper triangular matrix makes all subsequent entries in row 1 equal to 0 and makes the pivot element equal to 1. (ii) Premultiplication by a lower triangular matrix makes all elements below the pivot element equal to 0. (iii) Find a new pivot entry in row 2, and proceed recursively. (iv) At the end we are left with a permutation matrix P (since A is nonsingular) and so P = L0AU 0 where L0 is a lower triangular matrix (the product of all the lower triangular matrices used) and U 0 is an upper triangular matrix (the product of all the upper triangular matrices used). (v) Solving for A, we get A = LP U where L = (L0)−1 and U = (U 0)−1. We now show that the permutation matrix P is uniquely determined by A. Let P correspond to the permutation σ. Consider the initial parts of the rows of A, namely ri(A, j) = A[i | 1, 2, . . . , j], j = 1, 2, . . . , n, i = 1, 2, . . . , n, which denotes row i from column 1 to column j. For i = 1, 2, . . . , n, let ρ(A : i) = min j | ri(A, j) is not in the span of {r1(A, j), . . . , ri−1(A, j)} . (1.1) For P we obviously have that ρ(P : i) = σ(i), since the unique 1 in row i occurs in column σ(i). It is easy to verify that the value in (1.1) is preserved under multiplication on the left by a nonsingular lower triangular matrix L and on the right by a nonsingular upper triangular matrix U, that is, ρ(A : i) = ρ(LAU : i). Thus, P is uniquely determined by A. The second assertion follows from the first by reasons already discussed. Example 1.1.11. In this example, we show that the lower and upper triangular matrices in the matrix Bruhat decomposition are not necessarily unique. As before, 2 Qn denotes the anti-identity permutation matrix, where we have Qn = In. Let L1 be any n × n nonsingular lower triangular, and let U1 = QnL1Qn, a nonsingular upper triangular. Define an n×n A by A = L1QnU1, a matrix Bruhat 8 Chapter 1. Some Combinatorially Defined Matrix Classes decomposition. Then,

2 2 2 A = L1QnU1 = L1Qn(QnL1Qn) = L1(Qn)L1Qn = L1Qn = L1QnIn is another matrix Bruhat decomposition of A, diﬀerent in general from the ﬁrst since L1 was arbitrary. Note that the permutation matrix in the decomposition is the same.

We can now formalize the matrix Bruhat decomposition. Let GLn be the linear group of n × n nonsingular complex matrices: Then, GLn = LnSnLn, where Ln is the subgroup of (nonsingular) lower triangular matrices and Sn is the subgroup of n × n permutation matrices. Since the permutation matrix P is unique in the matrix Bruhat decomposition LP L of GLn, this gives a partition of GLn into double cosets parametrized by the n × n permutation matrices. Note that GLn can be replaced by a wider class of groups with Ln and Sn replaced by a wider class of subgroups; see Bj˝orner–Brenti [2].

1.1.5 Flags

Let Cn denote the n-dimensional complex vector space. Then a (complete) flag in Cn is a sequence of subspaces F : F0 = {0} ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn = Cn, where dim Fk = k for all k. Flags in Cn are equivalent to n × n nonsingular complex matrices A through the following correspondence: if the row vectors of A are v1, v2,..., vn then, letting Fk = hv1, v2,..., vki the linear span of v1, v2,..., vk, we have that v1, v2,..., vk is a basis of Fk and we obtain a flag of Cn. Of course, two different matrices may give rise to the same flag, since bases of vector spaces are not unique. But under the action of the linear group GLn, there is only one −1 −1 flag of Cn: since A is nonsingular, A exists and A · A = In, and thus every basis can be brought to the standard basis e1, e2,..., en by a nonsingular linear transformation. Thus, all flags of Cn become

{0} ⊂ < e1 > ⊂ < e1, e2 > ⊂ · · · ⊂ < e1, e2,..., en > .

(1) (2) We now consider pairs of flags F and F of Cn. These correspond to two n×n nonsingular complex matrices A1 and A2 with row vectors v1, v2,..., vn and (1) (2) w1, w2,..., wn, respectively: Fk = hv1, v2,..., vki and Fk = hw1, w2,..., wki, k = 0, 1, . . . , n. Under the action of GLn, there exists a nonsingular matrix C (in −1 fact, C = A2A1 ) such that A2 = CA1. Let the matrix Bruhat decomposition of C be C = L1PL2, where P is the permutation matrix corresponding to the −1 permutation σ. Then, A2 = CA1 = L1PL2A1 and thus, (L1 A2) = P (L2A1). Therefore, under a change of basis (determined by the nonsingular lower triangular −1 (1) (2) matrix L1 for F and by the nonsingular lower triangular matrix L2 for F ), the configuration type of the pair of flags F (1), F (2) is that of the pair of flags (1) (2) (1) (2) B , B , where Bk = hu1, u2,..., uki and Bk = huσ(1), uσ(2),..., uσ(k)i, k = 0, 1, . . . , n, for some basis u1, u2,..., un of Cn. Hence, configuration types of double 1.1. Permutations and Permutation Matrices 9

ﬂags of Cn are indexed by permutations σ. In fact, we may take u1, u2,..., un to be (1) (2) e1, e2,..., en, and hence Bk =he1, e2,..., eki and Bk =heσ(1), eσ(2),..., eσ(k)i, k = 0, 1, . . . , n. We have that

(1) (2) dim(Bi ∩ Bj ) = {1, 2, . . . , i} ∩ {σ(1), σ(2), . . . , σ(j)} , 1 ≤ i, j ≤ n.

Suppose that P is the n × n anti-identity matrix Qn corresponding to the permutation σ = (n, n − 1,..., 1). Then,

1 2 + dim(Bi ∩ Bj ) = {1, 2, . . . , i} ∩ {σ(1), σ(2), . . . , σ(j)} = (i + j − n) , (1.2)

+ the number of 1s of Qn in the leading i × j submatrix of Qn. (Here, a = max{a, 0}.) The intersection in (1.2) has the smallest possible dimension of the intersection of any i-dimensional subspace U and a j-dimensional subspace of Cn by virtue of the elementary inequality dim(U ∩ W ) ≥ (i + j − n)+. Thus, the pair of ﬂags B1, B2 given by

1 B : h0i ⊂ hu1i ⊂ hu1, u2i ⊂ · · · ⊂ hu1, u2,..., uni,

2 B : h0i ⊂ huni ⊂ hun, un−1i ⊂ · · · ⊂ hun, un−1,..., u1i,

n n where u1, u2,..., un is any basis of C , is the most generic pair of ﬂags in C . 1 2 In general, dim(Bi ∩ Bj ) = |{1, 2, . . . , i} ∩ {σ(1), σ(2), . . . , σ(j)}|, and the larger these intersections are, the less generic the ﬂag pair is. The set Pn of n × n permutation matrices is one instance of a collection of matrix classes. Let R = (r1, r2, . . . , rm) and S = (s1, s2, . . . , sn) be vectors with nonnegative integral entries, and let N (R,S) denote the class of all m × n nonnegative integral matrices whose row sums are given by R, and whose column sums are given by S. Thus, Pn is the special case with m = n and R = S = (1, 1,..., 1). The following well-known theorem gives a simple criterion in order that N (R,S) =6 ∅; see, e.g., Brualdi [5]. Theorem 1.1.12. The class N (R,S) is nonempty if and only if

r1 + r2 + ··· + rm = s1 + s2 + ··· + sn. (1.3)

Proof. The condition (1.3) is clearly a necessary condition for N (R,S) =6 ∅. Now assume that (1.3) holds. A simple recursive algorithm constructs a matrix A = [aij] in N (R,S):

(i) Choose any position (i, j) and set aij = min{ri, sj} where, if min{ri, sj} = ri, all other positions in row i are set equal to 0, and if min{ri, sj} = sj, all other positions in column j are set equal to 0.

(ii) Reduce ri and sj by aij (one of them gets reduced to 0 and the corresponding row or column gets deleted) to obtain new nonnegative integral vectors R0 and S0 satisfying the corresponding condition (1.3). 10 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

(iii) Proceed recursively. Example 1.1.13. Let R = (3, 5, 4) and S = (2, 4, 3, 3). Choosing the (1,1) position, then (1,2), then (2,2), (2,3), (3,3), and (3,4) we obtain the matrix

 2 1 0 0   0 3 2 0  0 0 1 3 in N (R,S).

The Bruhat order on Sn can be generalized to each class N (R,S). Let A1,A2 ∈ N (R,S). Using our previous notation, we write A1 B A2 if and only if A1 can be gotten from A2 by a sequence of moves of the form      a b   a + 1 b − 1            −→            c d   c − 1 d + 1 

1 −1 that is, by adding −1 1 to some 2 × 2 submatrix of A1, where b, c ≥ 1. These moves keep one in the class N (R,S), and it is immediate that

A1 B A2 =⇒ Σ(A1) ≥ Σ(A2).

As with the Bruhat order on Sn, we have the following theorem; see Magyar [25].

Theorem 1.1.14. If A1 and A2 are matrices in N (R,S), then A1 B A2 if and only if Σ(A1) ≥ Σ(A2) (entrywise). If in our algorithm to construct a matrix in N (R,S), we recursively choose the position in the northwest corner (so, starting with position (1, 1)), we obtain the unique minimal element in the Bruhat order on N (R,S). This is so because, by recursively choosing the northwest corner and inserting min{ri, sj} in that position, we are always obtaining the maximum value of σij(A) possible for a matrix in N (R,S). This is how the matrix in the Example 1.1.13 was constructed. Similarly, recursively choosing the position in the northeast corner (so starting with position (1, n)), we get the unique maximal element. In the case of Sn, this gives In and Qn, respectively. We now briefly extend our discussion of pairs of flags of Cn to pairs of partial n flags of C ; see Magyar [25]. Let b = (b1, b2, . . . , br) and c = (c1, c2, . . . , cs) be Pr Ps vectors of positive integers with i=1 bi = j=1 cj = n. Corresponding to b and n c, we have partial flags B : {0} = B0 ⊂ B1 ⊂ · · · ⊂ Br of subspaces of C with dim Bi/Bi−1 = bi, i = 1, 2, . . . , r, and C : {0} = C0 ⊂ C1 ⊂ · · · ⊂ Cq of subspaces n of C with dim Ci/Ci−1 = ci, i = 1, 2, . . . , s. The orbits FM of pairs of partial flags 1.1. Permutations and Permutation Matrices 11

Flag(b)×Flag(c) are indexed by r × s nonnegative integral matrices M = [mij] with row sum vector b and column sum vector c as follows: Take a basis of n vectors (vijk : 1 ≤ i ≤ r, 1 ≤ j ≤ s, 1 ≤ k ≤ mij) of n 0 C and let partial ﬂags B and C be deﬁned by Bi = hvi0jk | 1 ≤ i ≤ ii and 0 Cj = hvij0k | 1 ≤ j ≤ ji. As the basis varies we get the orbit FM . Note that the rank numbers are X rij(M) = dim(Bi ∩ Bj) = mkl. 1≤k≤i,1≤l≤j

The Bruhat order on nonnegative integral matrices (in its two equivalent characterizations) describes the degeneration order on the partial ﬂags.

1.1.6 Involutions and Symmetric Integral Matrices In this subsection we discuss another connection between permutations and integral matrices; see Brualdi–Ma [13]. 2 A permutation σ ∈ Sn is an involution provided σ = ιn. In terms of the 2 corresponding n×n permutation matrix P , this means P = In. The permutation matrix P is an involution if and only if P is a symmetric matrix. Example 1.1.15.  1   1     1  P =    1     1  1 is an involution whose corresponding permutation σ = (3, 6, 1, 4, 5, 2) has two ﬁxed points (corresponding to the two ones on the main diagonal).

Recall that an ascent of a permutation σ = (i1, i2, . . . , in) occurs at a position k when ik < ik+1, and a descent occurs at position k when ik > ik+1. The permutation σ in Example 1.1.15 has ascents at positions 1, 3, and 4 and descents at positions 2 and 5. An ascent in a permutation σ at a position k becomes a descent at position k in σ−1, so we just consider descents. Let I(n, k), k = 1, 2, . . . , n−1, equal the set of involutions of Sn with exactly k descents, and let I(n, k) = |I(n, k)|, 1 ≤ k ≤ n−1, Pn−1 k be their number with In(t) = k=1 I(n, k)t its generating polynomial.

Example 1.1.16. If n = 5, then there are 26 involutions in S5, and it can be veriﬁed 2 3 4 that I5(t) = 1 + 6t + 12t + 6t + t .

As in this example, the coeﬃcients of the generating polynomial In(t) are symmetric and unimodal but, in general, not log-convex (a stronger property than unimodality); see Barnabei–Bonetti–Silimbani [1]. 12 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

Let T (n, k) denote the set of k × k nonnegative integral, symmetric matrices without zero rows or columns whose sum of entries equals n, and let T (n, k) = |T (n, k)|. (Note well that n does not denote the size of the matrix; that size is k.) Example 1.1.17. The matrix  1 0 3 1   0 4 2 1     3 2 0 2  1 1 2 3 is in T (26, 4).

The following connection between sets of involutions I(n, k) in Sn (n×n symmetric permutation matrices) with k descents, whose cardinalities are the numbers I(n, k), and k × k nonnegative integral symmetric matrices with sum of entries equal to n, whose cardinalities are the numbers T (n, k), is from Brualdi–Ma [13]. Pn−1 k+1 n−1−k Pn i Theorem 1.1.18. k=0 I(n, k)t (1 + t) = i=1 T (n, i)t . Equivalently, i−1 X n − 1 − k T (n, i) = I(n, k) , i = 1, 2, . . . , n. i − 1 − k k=0 The equivalence of the two equations follows by expanding (1 + t)n−1−k using the binomial theorem. We shall outline a proof of Theorem 1.1.18 primarily through an example as given in [13]. Every position k with 1 ≤ k ≤ n − 1 in a permutation in Sn is either an ascent or descent. Thus, if a permutation has k descents, then it has (n − 1 − k) 0 ascents. Let I (n, k) denote the set of all permutations in Sn with exactly k ascents, and let I0(n, k) = |I0(n, k)| be their number (1 ≤ k ≤ n − 1). Then, I(n, k) = I0(n, n − 1 − k). Making this replacement in Theorem 1.1.18 and using the fact n−1−k n−1−k that i−1−k = n−i , we get

i−1 X n − 1 − k T (n, i) = I0(n, n − 1 − k) . n − i k=0 Finally, letting j = n − 1 − k, we get

n−1 X j T (n, i) = I0(n, j) , i = 1, 2, . . . , n. (1.4) n − i j=n−i

Equation (1.4) suggests that there may be a mapping Fn−i from the set of all n × n symmetric permutation matrices P with j ≥ n − i ascents onto the subsets of the set T (n, i) of i × i nonnegative integral symmetric matrices without zero j rows or columns whose entries sum to n, such that |Fn−i(P )| = n−i , where the Fn−i(P ) determines a partition of the set T (n, i). We illustrate such a mapping in the next example. 1.1. Permutations and Permutation Matrices 13

Example 1.1.19. Let n = 8 and consider the involution σ = (5, 7, 8, 6, 1, 4, 2, 3) with corresponding symmetric permutation matrix

 1   1     1     1  P =   .  1     1     1  1

Let i = 5 so that n − i = 3. We illustrate the mapping Fn−i = F3. There are j = 4 ascents and they occur in the pairs of positions (row indices of P ) {1, 2}, {2, 3}, {5, 6}, and {7, 8}. Choosing any subset of three of these pairs, for instance the pairs {1, 2}, {2, 3}, and {7, 8}, we obtain by combining consecutive pairs of ascents the partition U1 = {1, 2, 3}, U2 = {4}, U3 = {5}, U4 = {6}, U5 = {7, 8} of {1, 2,..., 8} with corresponding partition of the permutation matrix P into blocks given by

 1   1     1       1  P =   .  1       1     1  1

Adding the entries in each block gives the 5 × 5 nonnegative integral symmetric matrix without zero rows and columns whose sum of entries equals 8:

 0 0 1 0 2   0 0 0 1 0    A =  1 0 0 0 0  .    0 1 0 0 0  2 0 0 0 0

Next, to invert the mapping Fn−i, we need to know that if A is an i × i symmetric, nonnegative integral matrix with no zero rows and columns whose sum of entries equals n, then A results from exactly one n × n symmetric permutation matrix with at least n − i ascents in the above way. Let rk be the sum of the entries in row (and column) k of A, 1 ≤ k ≤ n. If A is to result from an n × n symmetric permutation matrix P by our procedure then, since P has exactly one 1 in each row and column, it must use the partition of the row and column indices 14 Chapter 1. Some Combinatorially Deﬁned Matrix Classes of P into the sets:

U1 = {1, . . . , r1},U2 = {r1 + 1, . . . , r1 + r2},...

...,Ui = {r1 + r2 + ··· + ri−1 + 1, r1 + r2 + ··· + ri}.

There must be a string of rk −1 consecutive ascents corresponding to the positions in each Uk. There may be ascents or descents in the remaining position pairs:

(r1, r1 + 1), (r1 + r2, r1 + r2 + 1),..., (r1 + r2 + ··· + ri−1, r1 + r2 + ··· + ri−1 + 1).

One needs to show that there is exactly one involution (symmetric permutation matrix) with these restrictions. We illustrate this in the next example.

Example 1.1.20. Let

 0 2 0  A =  2 1 3  0 3 0

(n = 11, r1 = 2, r2 = 6, r3 = 3). We seek an 11 × 11 symmetric permutation matrix of the form

 0 0 0 0 0   0 0 0 0 0                    ,            0 0 0 0 0     0 0 0 0 0  0 0 0 0 0 with one ascent in rows 1 and 2, ﬁve ascents in rows 3 to 8, and two ascents in rows 9, 10, and 11. Using the fact that we seek a symmetric permutation matrix P , it is easy to see that the only possibility for the two 1’s in rows 1 and 2, which must form an ascent, is the positions (1, 3) and (2, 4); otherwise, in rows 3, 4, 5, 6, 7, 8, we would have a pair of consecutive rows forming an ascent, a contradiction. Continuing to argue like this, we see that P must be the symmetric permutation matrix 1.2. Alternating Sign Matrices 15

 0 0 1 0 0 0   0 0 1 0 0 0     1       1     1     1  ,    1     1     0 0 1 0 0 0     0 0 1 0 0 0  0 0 1 0 0 0 equivalently, the involution 3, 4; 1, 2, 5, 9, 10, 11; 6, 7, 8. Notice that the pairs of positions which could be either ascents or descents, namely {2, 3} and {8, 9}, are both descents in this case.

1.2 Alternating Sign Matrices

An alternating sign matrix, abbreviated ASM, is an n × n (0, 1, −1)-matrix such that, ignoring 0’s, the 1’s and −1’s in each row and each column alternate, beginning and ending with a 1. ASMs are generalizations of permutation matrices, which are the ASMs without any −1’s. In this section we discuss the basic and important properties of ASMs and results from [3, 9, 14, 23]. We also discuss equivalent formulations of ASMs which give additional insight of their nature. Finally, we consider the extension of the Bruhat order on permutations to ASMs, a lattice which is the MacNeille completion [24] of the Bruhat order on permutations.

1.2.1 Basic Properties

Permutation matrices are the simplest examples of ASMs but there are many other ASMs which contain −1’s.

Example 1.2.1. The only 3 × 3 ASM which is not a permutation matrix is

 0 1 0  D3 =  1 −1 1  . 0 1 0

As with ASMs in general, we also write D3 using the notation

 0 + 0   +   + − +  or  + − +  . 0 + 0 + 16 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

Other examples are

 +     + − +  +    +   + − +       + − +  D5 =  + − + − +  and   .    + − + − +   + − +     +  +    +  +

Let An denote the set of all n × n ASMs. The ASM D5 ∈ A5 above has the largest number of nonzero elements in A5. We now list some readily verifiable basic properties of ASMs: (i) The first and last rows and columns contain a unique +1 and no −1. (ii) The number of 1’s in each row and column is odd. (iii) All row and column sums equal 1. More generally, the partial row (resp., column) sums starting from the first or last entry in a row (resp., column) equal 0 or 1. (iv) Let |A| be the (0, 1)-matrix obtained from an ASM A by replacing entries with their absolute values (so replacing −1’s with +1’s). The number of 1’s in the rows and columns of |A| is entrywise bounded by (1, 3, 5, 7,..., 7, 5, 3, 1). If n is odd, there is exactly one n × n ASM Dn such that |Dn| has these row and column sums (e.g., D5 above). If n is even, there are exactly two n × n 0 0 ASMs Dn and Dn such that |Dn| and |Dn| have these row and column sums 0 0 (e.g., D4 and D4 below). The ASMs Dn (n odd), and Dn and Dn (n even) are called diamond ASMs. They have the largest number of nonzeros of all ASMs in An. (v) Regarding an ASM as a square of numbers, an ASM is transformed into another ASM under the action of the dihedral group of symmetries of a square. Thus, for instance, under a rotation by 90 degrees, we have

 +   +   + − +   + − +  0 D4 =   −→   = D .  + − +   + − +  4 + +

1.2.2 Other Views of ASMs

Let A = [aij] be an n×n ASM. Let C(A) be the n×n (0, 1)-matrix whose i-th row is obtained by summing the entries in the columns of rows 1, 2, . . . , i of A; C(A) is the column sum matrix of the ASM A. Each row i of C(A) contains exactly i 1.2. Alternating Sign Matrices 17

1’s and n − i 0’s, since each row of an ASM contains one more 1 than −1 and thus the total sum of the entries in rows 1 to i is i. In particular, the last row of A is all 1’s. For example,

 1   0 0 0 1 0   1 −1 1   0 1 0 0 1      A =  1 −1 1  −→ C(A) =  1 0 1 0 1  .      1 −1 1   1 1 0 1 1  1 1 1 1 1 1

If the rows of C(A) are c1, c2,..., cn, then A is recovered from C(A) by   c1  c2 − c1    .  .   .  cn − cn−1

Another view of an ASM A is that of a monotone triangle T of order n deﬁned to be a triangular arrangement of n(n + 1)/2 integers tij taken from {1, 2, . . . , n} with shape 1, 2, . . . , n and such that

(i) tij < ti,j+1 for 1 ≤ j ≤ i (strictly increasing in rows), and

(ii) tij ≤ ti−1,j ≤ ti,j+1 for 1 ≤ j ≤ i − 1. For example, 3 2 4 1 3 4 1 2 3 4 is a monotone triangle of order 4. An ASM A has an associated monotone triangle T (A) of order n where the entries tij in row i are the column indices of the 1’s in row i of C(A). Thus, T (A) is just another way of specifying the (0, 1)-matrix C(A). That T (A) satisfies property (i) is obvious from the definition. For property (ii), if a 1 in row j − 1 occurs in column k, then either it remains a 1 in row j − 1, or else A has a −1 in position (j, k) which is then preceded by a 1, creating a new column sum of 1 in row j. Since row j can have −1’s only in those columns which sum to i up to column (j − 1), this establishes property (ii). Since T (A) is another way to describe C(A), this gives a characterization of the column sum matrices of ASMs. A third view of ASMs is that originating from earlier work by physicists; see Bressoud [3]. There is a 1-1 correspondence between ASMs and so-called “square ice” configurations described as a system of water (H2O) molecules frozen in a square lattice. 18 Chapter 1. Some Combinatorially Defined Matrix Classes

There are oxygen atoms at each vertex of an n × n lattice, with hydrogen atoms between successive oxygen atoms in a row or column, and on either vertical side of the lattice, but not on the two upper and lower horizontal sides. For example, with n = 4 we have: HOHOHOHOH HHHH HOHOHOHOH HHHH HOHOHOHOH HHHH HOHOHOHOH.

Each O is to be attached to two H’s (giving a water molecule H2O) in a one-to-two bijection. There are six possible conﬁgurations in which an oxygen atom can be attached to two hydrogen atoms: H ↑ H ← O → H O ↓

H H O → H H ← O ↑ ↑ ↓ ↓ H ← O O → H H H. Letting the horizontal configuration correspond to 1, the vertical configuration correspond to −1, and the other four (skew) configurations correspond to 0’s gives an ASM. Example 1.2.2. The following is a square ice configuration: H ← OH ← OH ← O → HO → H ↓ ↓ ↓ HHHH ↑ H ← O → HO → HOH ← O → H ↓ ↓ HHHH ↑ ↑ H ← OH ← O → HO → HO → H ↓ HHHH ↑ ↑ ↑ H ← OH ← OH ← O → HO → H 1.2. Alternating Sign Matrices 19 corresponding to the ASM  0 0 1 0   1 0 −1 1    .  0 1 0 0  0 0 1 0 This correspondence is reversible, giving a bijection between square ice configurations and ASMs.

1.2.3 The λ-determinant The λ-determinant arose in the work of Mills, Robbins, and Rumsey (see [3, 26]) as a generalization of the classical determinant of a square matrix. Its deﬁnition is based on Dodgson’s classical recursive formula for the determinant. Let A = [aij] be an n × n matrix. Let AUL, ALR, AUR, and ALL be, respectively, the (n − 1) × (n − 1) submatrices of A in the Upper Left corner, Lower Right corner, Upper Right corner, and Lower Left corner. Also let AC be the (n − 2) × (n − 2) submatrix of A in the middle (obtained by deleting the ﬁrst and last rows and columns). Then Dodgson’s formula is detA detA − detA detA detA = UL LR UR LL . detAC

Starting with the fact that the determinant of a 1 × 1 matrix A = [a11] equals a11, this formula enables one to calculate any determinant (at least symbolically). In the ﬁrst application for a 2 × 2 matrix, AC is an empty matrix with determinant equal to 1. To deﬁne the λ-determinant detλ(A) of an n × n matrix A, we start with detλ[a11] = a11 and, for n ≥ 2, use

detλAULdetλALR + λdetλAURdetλALL detλA = . detλAC If λ = −1, then, by Dodgson’s formula, we get the classical determinant. If n = 2, we get a11 a12 detλ = a11a22 + λa12a21. a21 a22 If n = 3, we get

2 −1 detλ(A) = a11a22a33 + λa12a21a33 + λa11a23a32 + (λ + λ)a12a21a22 a23a32 2 2 3 + λ a13a21a32 + λ a12a23a31 + λ a13a22a31.

There are seven terms in the λ-determinant of a 3 × 3 matrix. If λ = −1, the term with coeﬃcient (λ2 + λ) equals 0, and only six terms remain, and these give the classical determinant. 20 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

If for each of the seven terms in the λ-determinant of a 3 × 3 matrix, we construct a 3 × 3 matrix whose entry in its (i, j)-position is the exponent of aij in that term, we get seven matrices. Six of these are 3 × 3 permutation matrices, and the seventh is

 0 1 0  −1  1 −1 1  ←→ a12a21a22 a23a32. 0 1 0

More generally, if A = [aij] is an n × n matrix, then detλA is of the form

n X Y bij pB(λ) aij ,

B=[bij ]∈ASMn×n i.j=1 where pB(λ) is a polynomial in λ. Thus, the terms in the λ-determinant of an n × n matrix are indexed by the n × n alternating sign matrices. This fact led to the question of how many n × n ASMs there are. The number of n×n ASMs has been calclulated for n = 1, 2, 3, 4, 5, 6 and these are 1, 2, 7, 42, 429, and 7436. It was conjectured in Mills–Robbins–Rumsey [26] and proved ﬁrst in Zeilberger [28] and later in Kuperberg [21] that the number of n×n ASMs is given by 1!4!7! ··· (3n − 2)! . n!(n + 1)!(n + 2)! ··· (2n − 1)!

1.2.4 Maximal ASMs

An extension of an n × n ASM A = [aij] is another n × n ASM B = [bij] such that B =6 A and aij =6 0 implies bij = aij, for 1 ≤ i, j ≤ n. Thus, an extension of an ASM A is obtained by changing at least one nonzero into +1 or −1 (actually at least four changes are required to get another ASM). A maximal ASM is an ASM without any extensions.

Example 1.2.3. Identity matrices are maximal. For instance, the identity matrix I5 is maximal, since a 5 × 5 ASM with 1’s on the main diagonal cannot have any more nonzeros in rows and columns 1 and 5, and then cannot have nonzeros in rows and columns 2 and 4. The ASM

 0 1 0 0 0   0 0 0 1 0    A =  1 0 0 0 0     0 0 0 0 1  0 0 1 0 0 1.2. Alternating Sign Matrices 21 is not maximal since  0 1 0 0 0   0 0 0 1 0    B =  1 −1 1 0 0     0 1 −1 0 1  0 0 1 0 0 is an extension.

Let 1 ≤ i < j ≤ n and 1 ≤ k < l ≤ n, and let Tn(i, j; k, l) be the n × n (0, 1, −1)-matrix whose submatrix determined by rows i and j, and columns k and l, equals 1 −1 −1 1 with all other entries equal to zero. An elementary extension of an ASM A is an ASM B that results from A by adding one of the matrices Tn(i, j : k, l) to A, where the nonzero positions of Tn(i, j; k, l) are zero positions in A. The matrix B in the preceding example is an elementary extension of A where B = A−T5(3, 4; 2, 3). It can be shown that a permutation matrix is not maximal if and only if it has an elementary extension; see Brualdi–Kiernan–Meyer–Schroeder [9].

1.2.5 Generation

It is possible to add a matrix Tn(i, j : k, l) to an ASM A, where Tn(i, j; k, l) and A have at least one overlapping nonzero position with the result being an ASM. A simple example is the matrix B in Example 1.2.3, which satisﬁes B+T5(3, 4; 2, 3) = A with four overlapping positions. Example 1.2.4. Let

 0 0 +1 0 0   0 +1 0 0 0    A =  +1 0 −1 0 +1  .    0 0 0 +1 0  0 0 +1 0 0

Then,  0 0 +1 0 0   0 +1 −1 +1 0    A − T5(2, 3; 3, 4) =  +1 0 0 −1 +1     0 0 0 +1 0  0 0 +1 0 0 is an ASM with only two more nonzero positions than A. Let A be an n×n ASM. Deﬁne an ASM-interchange of A to be the operation of adding one of the matrices Tn(i, j; k, l) to A resulting in another ASM B. If A 22 Chapter 1. Some Combinatorially Deﬁned Matrix Classes and B are both permutation matrices, then the ASM-interchange corresponds to the classical notion of a transposition of a permutation. Example 1.2.5. Let  1   1    A =  1  .    1  1 Then,  1   1    A + T5(1, 3; 4, 5) =  1  ,    1  1 and this corresponds to the transposition (4, 2, 5, 1, 3) → (5, 2, 4, 1, 3), in which 4 and 5 have changed positions. In Brualdi–Kiernan–Meyer–Schroeder [9] it is proved that any n×n ASM can be gotten from the identity matrix In by a sequence of ASM-interchanges. Since ASM-interchanges are reversible, if A1 and A2 are any two n × n ASMs, A2 can be obtained from A1 by a sequence of ASM-interchanges. This fact also follows from the work in Lascoux–Sch˝utzenberger [23] discussed in the next section.

1.2.6 MacNeille Completion and the Bruhat Order The following theorem is due to MacNeille [24]. Recall that a ﬁnite lattice is a ﬁnite partially ordered set in which every pair {a, b} of distinct elements has a unique greatest lower bound a ∧ b, and a unique least upper bound a ∨ b.

Theorem 1.2.6. Let (P, ≤1) be a ﬁnite partially ordered set. Then, there exists a unique minimal lattice (L, ≤2) such that P ⊆ L and, for a, b ∈ P , a ≤1 b if and only if a ≤2 b.

The lattice (L, ≤2) in Theorem 1.2.6 is called the MacNeille completion of (P, ≤1). Lascoux–Sch˝utzenberger [23] determined the MacNeille completion of (Sn, B); see also a generalization in Fortin [18].

Theorem 1.2.7. The MacNeille completion of (Sn, B) is (An, B), where B in (Pn, B) is the Bruhat order on An deﬁned by

A1 B A2 if and only if Σ(A1) ≥ Σ(A2)(entrywise).

Thus, the elements of the MacNeille completion of the n × n permutation matrices with the Bruhat partial order are the n × n alternating sign matrices. 1.2. Alternating Sign Matrices 23

 0 0 1  L3 =  0 1 0  1 0 0  0 1 0   0 0 1   0 0 1   1 0 0  1 0 0 0 1 0  0 1 0  D3 where D3 =  1 −1 1  0 1 0  0 1 0   1 0 0   1 0 0   0 0 1  0 0 1 0 1 0  1 0 0  I3 =  0 1 0  0 0 1

Figure 1.2: Hasse diagram of (A3, B).

The Hasse diagram of (A3, B) is given in Fig. 1.2

The Hasse diagram of (A4, B) is given in Fig. 1.3; see Brualdi–Schroeder [14]. As a final remark, we mention that in Brualdi–Kim [12] a generalization of alternating sign matrices is investigated whereby instead of requiring that the first and last nonzero entries in each row and column equal +1, the first and last nonzero entries in each row and column are independently prescribed. This can be done by bordering an n × n array with 1 entries and then requiring that the signs of nonzero entries in the bordered matrix alternate in rows and columns. Such generalized ASMs are no longer guaranteed to exist.

Example 1.2.8. For a 3 × 3 matrix, suppose we consider the bordered array

+1 −1 +1 −1 ∗ ∗ ∗ +1 +1 ∗ ∗ ∗ +1 . −1 ∗ ∗ ∗ −1 −1 +1 −1 24 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

4321

3421 4231 4312

41 3412 14

0 2431 3241 D4 4132 4213

24 2341 42 13 4123 31

3142 33 23 32 22 2413

1432 34 43 12 21 3214

1342 1423 D4 2314 3124

11 2143 44

1243 1324 2134

1234

Figure 1.3: Hasse diagram of (A4, B). 1.2. Alternating Sign Matrices 25

Then, +1 −1 +1 −1 0 +1 −1 +1 +1 −1 0 0 +1 −1 +1 −1 +1 −1 −1 +1 −1

(the middle 3 × 3 matrix) is a generalized ASM for this border of 1’s. For a 2 × 2 matrix, suppose we consider the bordered array

+1 +1 +1 ∗ ∗ +1 . +1 ∗ ∗ −1 −1 +1

Then there is no way to replace the ∗’s by 0, +1, or −1 so that the signs of nonzero entries alternate in rows and columns in the bordered array.

Necessary and suﬃcient conditions for the existence of these generalized ASMs are given in Brualdi–Kim [12].

1.2.7 Bruhat Order Revisited

The entries of an ASM are 0’s, +1’s, and −1’s. The number of −1’s in an n × n ASM is determined by the number of +1’s, the number of +1’s is determined by the number of −1’s. But the positions of the −1’s in the ﬁrst instance and the +1’s in the second instance are not in general uniquely determined. In fact, if we begin with a (∗, −1)-matrix (or a (∗, +1)-matrix) X, we may not be able to replace some of the ∗’s with +1’s (or some of the ∗’s with −1’s) and obtain an ASM. When it is possible, we call the resulting matrix an ASM-completion of X. If an ASM-completion of an n × n (∗, −1) matrix (resp., (∗, +1)-matrix) is possible, then:

(i) there cannot be any −1’s in rows and columns 1 and n (resp., there must be exactly one +1 in each of rows and columns 1 and n);

(ii) there cannot be two consecutive −1’s in a row or column (resp., there cannot be two consecutive +1’s in a row or column); and

(iii) if U = (u1, u2, . . . , un) and V = (v1, v2, . . . , vn) record the number of −1’s in the rows and columns of A (resp., the number of +1’s) then, entrywise, U, V ≤ (0, 1, 2, 3,..., 3, 2, 1, 0) (resp., U, V ≤ (1, 2, 3,..., 3, 2, 1)). 26 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

Example 1.2.9. Let

 ∗ ∗ ∗ ∗ ∗ ∗ ∗   ∗ ∗ ∗ −1 ∗ ∗ ∗     ∗ ∗ −1 ∗ −1 ∗ ∗    X =  ∗ −1 ∗ ∗ ∗ −1 ∗  .    ∗ ∗ −1 ∗ −1 ∗ ∗     ∗ ∗ ∗ −1 ∗ ∗ ∗  ∗ ∗ ∗ ∗ ∗ ∗ ∗

Any completion of X must include +1’s in the positions as shown in

 +1   +1 −1 +1     +1 −1 −1 +1     +1 −1 −1 +1  .    +1 −1 −1 +1     +1 −1 +1  +1

(Here, as usual, unspecified positions are assumed to be 0’s.) But now, for an ASM completion, +1’s are required between four pairs of −1’s, but the result is never an ASM, since we are not allowed to insert −1’s. Note that if there had been a −1 in the middle position of X, then there would be a unique completion to an ASM, namely,  +1   +1 −1 +1     +1 −1 +1 −1 +1     +1 −1 +1 −1 +1 −1 +1  .    +1 −1 +1 −1 +1     +1 −1 +1  +1 The problem of ASM-completions of (∗, −1)-matrices and that of (∗, +1)- matrices are similar and we focus on (∗, −1)-matrices; see Brualdi–Kim [11, 10]. Let X = [xij] be an n × n (∗, −1)-matrix without any −1’s in rows and columns 1 and n. We associate with X a hypergraph H(X) whose vertices are the ∗-positions of X and whose edges are either horizontal edges or vertical edges. The set H = {H1,H2,...,Hp} of horizontal edges is the set of ∗-positions in rows to the left of the first −1 in a row, between succeeding −1’s in a row, and to the right of the last −1 in a row. The set V = {V1,V2,...,Vq} of vertical edges is defined in an analgous way using columns. The following theorem is now obvious. Theorem 1.2.10. The n × n (∗, −1)-matrix X has an ASM-completion if and only if the number p of horizontal edges of H(X) equals its number q of vertical edges, 1.2. Alternating Sign Matrices 27

and there is a permutation (i1, i2, . . . , ip) of {1, 2, . . . , p} such that H1 ∩ Vi1 =6 ∅,

H2 ∩ Vi2 =6 ∅, ..., Hp ∩ Vip =6 ∅.

Since a horizontal edge can intersect a vertical edge in at most one position, the positions of an ASM-completion of A with +1’s are the positions in

H1 ∩ Vi1 , H2 ∩ Vi2 , ..., Hp ∩ Vip . The number of ASM-completions of the (∗, −1)- matrix X equals the number of permutations (i1, i2, . . . , in) satisfying the conclusion of the theorem. In other words, we have a bipartite graph B(X) with vertices h1, h2, . . . , hp and v1, v2, . . . , vp with an edge between hi and vj if and only if Hi ∩ Vj =6 ∅. The (∗, −1)-matrix X has an ASM-completion if and only if B(X) has a perfect matching. Since perfect matchings in a bipartite graph can be found in polynomial time, there is a polynomial algorithm to determine an ASM-completion of an n × n (0, −1)-matrix. More generally, if A is the n × n bi-adjacency matrix of B(X), then the number of ASM-completions of X is the number of perfect matchings of B(X), and this equals the permanent of A. We now consider a special class of (0, −1)-matrices about which we can say something more; see Brualdi–Kim [11]. An n × n (0, −1)-matrix is a bordered- permutation (0, −1)-matrix provided that

(i) the ﬁrst and last rows and columns contain only zeros; and

(ii) the middle (n−2)×(n−2) submatrix is A[{2, 3, . . . , n−1} | {2, 3, . . . , n−1}] = −P , where P is a permutation matrix.

Example 1.2.11. A bordered-permutation (0, −1)-matrix and its completion to an ASM is illustrated below:

   +1   −1   +1 −1 +1      X =  −1  →  +1 −1 +1  .      −1   +1 −1 +1  +1

In this example, the set of vertices of the hypergraph H(X) is

(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 4), (2, 5), (3, 1), (3, 2), (3, 3), (3, 5), (4, 1), (4, 3), (4, 4), (4, 5), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5) .

The horizontal edges are H1 = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5) ,H2 = (2, 1), (2, 2) , H3 = (2, 4), (2, 5) ,H4 = (3, 1), (3, 2), (3, 3) ,H5 = (3, 5) ,H6 = (4, 1)}, H7 = (4, 3), (4, 4), (4, 5) ,H8 = (5, 1), (5, 2), (5, 3), (5, 4), (5, 5) . 28 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

The vertical edges are V1 = (1, 1), (2, 1), (3, 1), (4, 1), (5, 1) ,V2 = (1, 2), (2, 2), (3, 2) ,V3 = (5, 2) , V4 = (1, 3) ,V5 = (3, 3), (4, 3), (5, 3)},V6 = {(1, 4), (2, 4)},V7 = {(4, 4), (5, 4) , V8 = (1, 5), (2, 5), (3, 5), (4, 5), (5, 5) .

The placement of the +1’s corresponds to the permutation (4, 2, 6, 5, 8, 1, 7, 3):

H1 ∩ V4 = {(1, 3)},H2 ∩ V2 = {(2, 2)},H3 ∩ V6 = {(2, 4)},H4 ∩ V5 = {(3, 3)},

H5 ∩ V8 = {(3, 5)},H6 ∩ V1 = {(4, 1)},H7 ∩ V7 = {(4, 4)},H8 ∩ V3 = {(5, 2)}.

The following theorem is proved in Brualdi–Kim [11]: Theorem 1.2.12. Let n ≥ 2. Every n × n bordered-permutation (0, −1)-matrix has a completion to an ASM. If one deletes some of the −1’s from a bordered-permutation (0, −1)-matrix, then it is easy to say that an ASM-completion still exists. However, adding an additional −1 to a bordered-permutation (0, −1)-matrix may result in a matrix that is not completable to an ASM. Example 1.2.13. The matrix    −1     −1 −1     −1     −1     −1  is not completable to an ASM, since any such completion must include +1’s as shown in  +1   +1 −1 +1     +1 −1 +1 −1     +1 −1 +1  .    +1 −1 +1     +1 −1 +1  +1 This is not an ASM but no further +1’s are possible. Some bordered-permutation (0, −1)-matrices have unique ASM-completions, and these are characterized in the next theorem. An n × n bordered-permutation 1.2. Alternating Sign Matrices 29

(0, −1)-matrix X has a monotone decomposition provided X contains a position with −1 (a central position or central −1) that partitions A as  

 A11 A12           −1  , (1.5)        A21 A22 

where the −1’s in A11 and A22 are monotone decreasing by rows and those in A12 and A21 are monotone increasing by columns (so, e.g., the −1’s in A11, A12, and the central −1 occur in a set of positions which is “concave up”). Example 1.2.14. Here we give a monotone decomposition of a 9 × 9 bordered- permutation (0, −1)-matrix:    −1     −1     −1      X =  −1  .    −1     −1     −1 

The −1 in position (5, 6) is the central −1 of the monotone decomposition. X has a unique ASM completion which is

 +1   +1 −1 +1     +1 −1 +1     +1 −1 +1       +1 −1 +1  .    +1 −1 +1     +1 −1 +1     +1 −1 +1  +1

Theorem 1.2.15. Let n ≥ 3. An n × n bordered-permutation (0, −1)-matrix X has a unique completion to an ASM if and only if A has a monotone decomposition. 30 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

In contrast to Theorem 1.2.15, some bordered-permutation (0, −1)-matrices may have a substantial number of ASM-completions. In Brualdi–Kim [11], the following conjecture is made.

Conjecture. Let Em be the m × m (0, −1)-matrix with m − 1 −1’s in positions (2, m), (3, m − 1),..., (m, 2) and let Fm be the m × m (0, −1)-matrix with m − 1 −1’s in positions (1, m − 1), (2, m − 2),..., (m − 1, 1). If n ≥ 4 is even, then En/2 ⊕ Fn/2 is the n × n bordered-permutation (0, −1)-matrix with the largest number of ASM completions. If n ≥ 5 is odd, then E(n−1)/2 ⊕ (−I1) ⊕ F(n−1)/2 is the n × n bordered-permutation (0, −1)-matrix with the largest number of ASM completions.

Example 1.2.16. Let n = 10 and

   −1     −1     −1     −1  E5 ⊕ F5 =   .  −1     −1     −1     −1 

Every ASM-completion of E5 ⊕ F5 must have +1’s as shown in

 +1   +1 −1     +1 −1     +1 −1     +1 −1    .  −1 +1     −1 +1     −1 +1     −1 +1  +1

For the remainder of the completion, we need to put a permutation matrix of +1’s in the middle 8 × 8 submatrix. Thus, the number of completions to an ASM of 1.2. Alternating Sign Matrices 31

E5 ⊕ F5 is the permanent of the 8 × 8 matrix

 1 1 1 1   1 1 1 1 1     1 1 1 1 1 1     1 1 1 1 1 1 1    ,  1 1 1 1 1 1 1     1 1 1 1 1 1     1 1 1 1 1  1 1 1 1 and the conjecture asserts that this matrix has the largest permanent among the permanents of all the adjacency matrices of the bipartite graphs B(X), where A is a 10 × 10 bordered-permutation (0, −1)-matrix. More generally, in the even case, it asserts that the largest permanent3 is obtained by the (n − 2) × (n − 2) (0, 1)-Hankel matrix with an equal number of bands of 1’s to the left and below the upper right corner.

1.2.8 Spectral Radius of ASMs

We close this section on ASMs by brieﬂy discussing the spectrum of an ASM; see Brualdi–Cooper [6]. Let ρ(A) be the spectral radius of a square matrix A, that is, the maximum absolute value of an eigenvalue of X. According to the classical Perron–Frobenius theory, if A is a nonnegative matrix, then ρ(A) is an eigenvalue of A and has an associated nonnegative eigenvector. Now suppose that A is an n × n ASM. Since all row sums of A equal 1, it follows that the vector e = (1, 1,..., 1) of n 1’s is an eigenvector of A with eigenvalue 1. Hence ρ(A) ≥ 1. Equality holds for all ASMs that are permutation matrices, and can hold for other ASMs as well.

Example 1.2.17. Let A be the ASM

 0 0 +1 0 0   +1 0 −1 +1 0    A =  0 0 +1 −1 +1  .    0 0 0 +1 0  0 +1 0 0 0

2 2 Then, the characteristic polynomial√ of A is (λ − 1) (λ + 1)(λ − λ + 1) and, hence, A has eigenvalues 1, 1, −1, (1 −3)/2, all of which have absolute values equal to 1. Therefore ρ(A) = 1. We can also conclude that ρ(A) = 1 by observing that 6 A = I6.

3 P Recall that the permanent of an n n matrix B = [bij ] is b1i b2i bni . × (i1,i2,...,in)∈Sn 1 2 ··· n 32 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

Let ρn = max{ρ(A) | A ∈ A} be the maximum spectral radius of an n × n ASM. By the above, ρn ≥ 1. For a matrix X = [xij], let |X| = [|xij|] be the matrix obtained from X by replacing each entry with its absolute value. The diamond ASM Dn is signature similar to |Dn|, and hence its spectrum is the same as the spectrum of the nonnegative matrix |Dn|. In fact, let En be the n × n diagonal matrix −1 diag(1, −1, 1, −1, 1, −1,...). Then, En =En and EnDnEn = |Dn|. Thus, ρ(Dn)= ρ(|Dn|). Note that not every ASM has the same spectrum as its absolute value: Example 1.2.18. Let  0 0 +1 0   +1 0 −1 +1  A =   .  0 0 +1 0  0 +1 0 0 Then A is not signature similar to its absolute value

 0 0 1 0   1 0 1 1  |A| =   .  0 0 1 0  0 1 0 0

In fact, A and |A| have the same eigenvalues, namely 0, 1, 1, −1, but they have diﬀerent Jordan canonical forms.

Theorem 1.2.19. We have ρn = ρ(Dn). Moreover, if A is an n × n ASM, then ρ(A) = ρn if and only if A = Dn. Moreover, ρn = 2n/π + O(1).

The spectrum of the diamond ASM Dn has been investigated in Catral–Lin– Olesky–van-den-Driessche [15].

1.3 Tournaments and Tournament Matrices

A tournament Tn of order n is an orientation of the complete graph Kn of order n with vertex set {1, 2, . . . , n}. That is, for each unordered pair {p, q} of distinct integers from {1, 2, . . . , n}, one chooses a ﬁrst element and a second element, creating an ordered pair, e.g., (q, p), which is usually denoted as q → p. A tournament Tn is a model for a round-robin competition in which there are n teams (or players) and every pair of teams plays a game resulting in a winner and a loser. An arrow q → p indicates that team q beats team p. The score vector of Tn is Rn = (r1, r2, . . . , rn), where ri is the outdegree of vertex i. Thus, the score vector Rn records the number of wins of each team. The loss vector is Sn = (s1, s2, . . . , sn), where si is the indegree of vertex i and, thus, records the number of losses of team i. Since each team plays n − 1 games we have ri + si = n − 1 for each i. This section is based on Brualdi–Fritscher [7, 8]. 1.3. Tournaments and Tournament Matrices 33

Example 1.3.1. Here, we give an example of an orientation of K6 to produce a tournament T6 with score vector R6 = (4, 3, 3, 2, 2, 1):

4 5

1 6

2 3

For instance, 1 → 2, 1 → 3, 1 → 4, 1 → 6, but 5 → 1, giving a score of 4 for vertex 1 (with one loss).

A tournament of order n has an n × n adjacency matrix [tij], where tii = 0 for all i, and for i =6 j, tij = 1 and tji = 0 if and only if team i beats team j. We often do not distinguish between a tournament (as an orientation of Kn) and its corresponding adjacency matrix, called a tournament matrix. We refer to both as tournaments and denote both by Tn. Thus, for the tournament in Example 1.3.1, we have  0 1 1 1 0 1   0 0 0 1 1 1     0 1 0 0 1 1  T6 =   .  0 0 1 0 1 0     1 0 0 0 0 1  0 0 0 1 0 0

The score vector of a tournament Tn is the row sum vector Rn = (r1, r2, . . . , rn) of its corresponding tournament matrix, and its loss vector is the column sum vector S = (s1, s2, . . . , sn). Since we are free to label the vertices in any way we t wish (changing the labeling replaces the tournament matrix with PTnP for some n × n permutation matrix P ), there is no loss of generality in assuming that Rn is nondecreasing, that is, r1 ≤ r2 ≤ · · · ≤ rn.

1.3.1 The Inverse Problem

A tournament Tn has a score vector Rn consisting of n nonnegative integers. Since k each subset of k teams plays 2 games just among themselves, the sum of any k k k scores is at least 2 (equivalently, the sum of the k smallest scores is at least 2 ) n with the sum of all the scores equal to 2 . The inverse problem is the following: 34 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

given a vector Rn of n nonnegative integers, is there a tournament with score vector Rn? The answer is that the above necessary conditions are suﬃcient, and this result is usually called Landau’s theorem [22].

Theorem 1.3.2. Let Rn = (r1, r2, . . . , rn) be a vector of nonnegative integers with r1 ≤ r2 ≤ · · · ≤ rn. Then, Rn is the score vector of a tournament if and only if

k X k ri ≥ , (1.6) 2 i=1 for k = 1, 2, . . . , n, and with equality for k = n.

Let T (Rn) denote the set of all tournaments (tournament matrices) with score vector Rn. Theorem 1.3.2 determines when T (Rn) =6 ∅. In the next section we describe how one can obtain all tournaments in T (Rn) from one of them.

1.3.2 Generation

Let Rn = (r1, r2, . . . , rn) be a vector of nonnegative integers with r1 ≤ r2 ≤ · · · ≤ rn such that T (Rn) =6 ∅, and let Tn be any tournament in T (Rn). There are certain transformations which, when applied to Tn, give another tournament in T (Rn). Each of them replaces a 3 × 3 or 4 × 4 principal submatrix of Tn with another. These are shown below: (i) Type I: A transformation reversing the directions of the edges of a 3-cycle, equivalently, by the sequential interchange of 2×2 submatrices in the shaded regions of the principal 3 × 3 submatrices of the form C shown:

 0 1 0   0 0 1  0 C =  0 0 1  ←→  1 0 0  = C . 1 0 0 0 1 0

Note that after the ﬁrst interchange in the upper-right 2 × 2 submatrix, we no longer have a tournament matrix since a 1 is on the main diagonal. This 1 is then used in the second interchange in the lower-left 2 × 2 submatrix. These two interchanges have to be regarded as one transformation producing another tournament with the same score vector. (ii) Type II: A transformation reversing the directions of the edges of two 3- cycles, equivalently, by the simultaneous interchange of the 2×2 submatrices in the shaded regions of the 4 × 4 principal submatrices shown:

 0 a 1 0   0 a 0 1   1 − a 0 0 1   1 − a 0 1 0  0 D1 =   ←→   = D  0 1 0 b   1 0 0 b  1 1 0 1 − b 0 0 1 1 − b 0 1.3. Tournaments and Tournament Matrices 35

and  0 1 0 a   0 0 1 a   0 0 b 1   1 0 b 0  0 D2 =   ←→   = D .  1 1 − b 0 0   0 1 − b 0 1  2 1 − a 0 1 0 1 − a 1 0 0

Note that the 3-cycles reversed depend on the values of a and b.

Theorem 1.3.3. Given T1,T2 ∈ T (R), there is a sequence of transformations of Types I and II which transform T1 into T2. Put another way, any tournament with score vector R can be transformed into any other by reversing 3-cycles.

In the deﬁnition of a tournament the complete graph Kn can be replaced by any multigraph in which two distinct vertices may be joined by more than one edge (or no edges at all). Let G be a multigraph with vertex set {1, 2, . . . , n}.A multi-tournament Tn (based on G) is an orientation of G. Let C = [cij] be the adjacency matrix of G, that is, the n × n symmetric matrix with zero diagonal, where cij is the nonnegative integer equal to the number of edges joining vertices i and j, for i diﬀerent from j. Of the cij edges joining a pair of vertices {i, j}, some of them may be oriented from i to j and the rest from j to i. Let Rn = (r1, r2, . . . , rn) be the outdegree vector of Tn so that ri is the number of edges with initial vertex i, 1 ≤ i ≤ n. Thus, the adjacency matrix of Tn is a nonnegative integral matrix t A = [aij] such that A + A = C. We call such a multitournament a C-tournament and, as in the case of tournaments, we also identify a C-tournament with its adjacency matrix. If C = Jn − In, a C-tournament is an ordinary tournament. As with ordinary tournaments, the row sum vector Rn of Tn is the score vector of the Tn, and the column sum vector is the loss vector. Example 1.3.4. Let  0 5 1  C =  5 0 2  . 1 2 0 Then,  0 2 1  A =  3 0 1  0 1 0 is (the adjacency matrix of) a C-tournament with score vector R = (3, 4, 1). The loss vector is the vector S = (3, 3, 2). The following theorem from Hakimi [19] generalizes Landau’s Theorem 1.3.2, although Hakimi makes no reference to Landau. This theorem was also proved by Cruse [16], who was unaware of Hakimi’s theorem; see also Entringer–Tolman [17].

Theorem 1.3.5. Let C = [cij] be an n × n symmetric, nonnegative integral matrix with zeros on the main diagonal. A vector R = (r1, r2, . . . , rn) of nonnegative 36 Chapter 1. Some Combinatorially Deﬁned Matrix Classes integers is the score vector of a C-tournament if and only if X X r(J) = rj ≥ cij = c(J), (1.7) j∈J i,j∈J,i

1.3.3 Loopy Tournaments and Their Generation While it makes sense from the round-robin competition point of view to have zeros on the main diagonal of a tournament, from the matrix point of view there is no reason to require zeros on the main diagonal. We can incorporate ones on the main diagonal of an n × n tournament T = [tij] in the following way. Before the competition begins, each player i ﬂips a coin after calling heads or tails. If player i calls the coin correctly, the player gets a win, and we set tii = 1; otherwise, we set t tii = 0. In this way we get a loopy tournament T satisfying T +T = Jn −In +Dn, where Dn is an n × n (0, 2)-diagonal matrix. If Dn = 0, then T is an ordinary tournament (all the calls of the coin tosses are wrong!); if Dn = 2In, then all the calls are correct. A correct call in the coin toss adds 1 to a player’s score. As before, the score vector R = (r1, r2, . . . , rn) equals the number of 1’s in each row of the resulting loopy tournament T . The score vector still determines the losing vector S = (s1, s2, . . . , sn) since ri + si = n for all i. But now S is not in general 0 0 0 0 the vector of column sums of T . If S = (s1, s2, . . . , sn) is the column sum vector 0 of T , then ri + si = n − 1 or n + 1 depending on whether tii = 0 or tii = 1. Example 1.3.6. An example of a 5 × 5 loopy tournament is  0 1 1 0 0   0 1 1 1 1   0 1 1 0 1   1 2 1 1 1    t   T =  0 0 0 1 1  , where T + T =  1 1 0 1 1  .      1 1 0 0 1   1 1 1 0 1  1 0 0 0 1 1 1 1 1 2 1.3. Tournaments and Tournament Matrices 37

The score vector of T is (2, 3, 2, 3, 2); the losing vector is (3, 2, 3, 2, 3). The column sum vector of T is (2, 3, 2, 1, 4). In Theorem 1.3.2 it was assumed without loss of generality that the vector R = (r1, r2, . . . , rn) was nondecreasing. This can be weakened in the following way. Let k be a nonnegative integer. A vector R = (r1, r2, . . . , rn) is k-nearly nondecreasing provided there is a vector u = (u1, u2, . . . , un) such that ui ∈ {0, 1, . . . , k} with R − u nondecreasing. Thus, R is k-nearly nondecreasing if and only if rj ≥ ri − k for 1 ≤ i < j ≤ n. If k = 0, then R is nondecreasing. If k = 1, then we refer to nearly nondecreasing instead of k-nearly nondecreasing. For example, (3, 2, 3, 4, 3, 4) is nearly nondecreasing, as is seen by taking u = (1, 0, 0, 1, 0, 0), giving R − u = (2, 2, 3, 3, 3, 4).

Lemma 1.3.7. Let R = (r1, r2, . . . , rn) be a 2-nearly nondecreasing vector of non- Pk negative integers. Assume that R satisfies Landau’s conditions (1.6): i=1 ri ≥ k 2 , for k = 1, 2, . . . , n, and with equality if k = n. Then, the nondecreasing re- arrangement of R also satisfies Landau’s inequalities. The proof of Lemma 1.3.7 is by algebraic manipulation. As a consequence of Lemma 1.3.7, it suffices to assume R is 2-nearly nondecreasing in Landau’s theorem. Lemma 1.3.7 and Theorem 1.3.2 provide the following characterization of the row sum vectors of loopy tournaments.

Theorem 1.3.8. Let R = (r1, r2, . . . , rn) be a vector of nonnegative integers with r1 ≤ r2 ≤ · · · ≤ rn. Then, there exists a loopy tournament with score vector R if Pk k + and only if there is an integer t with 0 ≤ t ≤ n such that i=1 ri ≥ 2 + (k − t) , for k = 1, 2, . . . , n, and with equality when k = n. When these conditions are satisfied, the number of ones on the main diagonal of the loopy tournament is n − t, and the ones can be taken to be in the last (n − t) positions on the main diagonal (i.e., there is a loopy tournament with score vector R in which the “best” teams call the coin correctly). The proof of the sufficiency of these conditions follows by subtracting one from the n − t largest ri’s, yielding a nearly nondecreasing sequence, and then using the strengthening of Landau’s theorem provided by Lemma 1.3.7 to get an ordinary tournament. Finally, replacing the last n − t zeros on its main diagonal with ones gives the desired loopy tournament. Let T `(R) denote the set of all loopy tournaments with a score vector R. By the above, to construct a tournament in T `(R), we can subtract one from the largest n − t components of R, giving a nearly nondecreasing R0 which satisfies Landau’s conditions, construct a tournament in T (R0), and then change the zeros in the last n − t positions on the main diagonal to ones. There is also a bijection between T `(R) and a class of ordinary tournaments obtained as follows. (∗) ` Let R = (t, r1, r2, . . . , rn). We define the following bijection between T (R) and T (R(∗)): given T ∈ T `(R), then T ∗ ∈ T (R(∗)) is obtained from T by horizon- tally moving the entries (n−t 1’s and t 0’s) on the main diagonal to a new column 38 Chapter 1. Some Combinatorially Defined Matrix Classes

0, vertically moving one minus the entries on the main diagonal to a new row 0 (giving a row sum of t), and putting zeros everywhere on the main diagonal of the resulting (n + 1) × (n + 1) matrix. This mapping is easily seen to be reversible. This gives rise to the following alternative interpretation of a loopy tournament as suggested by Kirkland [20]: suppose a round robin tournament with n + 1 players has been completed but one of the teams (say Team 0) has subsequently been disqualiﬁed. In order to take into account the results of the games played with Team 0, a 1 (resp., a 0) is put on the i-th main diagonal of T if Team i beat (resp., loses to) Team 0. Example 1.3.9. An example of the bijection is:  0 1 0 1 0 1   0 1 0 1 0   0 0 1 0 1 0   0 1 0 1 1       1 0 0 0 1 1   1 1 0 1 0  −→   .    0 1 1 0 1 0   0 0 0 1 1    1 0 1 0 0  1 0 0 0 0 1  0 1 0 1 0 0

To determine how to generate all loopy tournaments in a class T `(R), we use the 3-cycle reversals for generating T (R∗) and the bijection between T `(R) and T (R∗). This gives that T `(R) can be generated starting from any T ∈ T `(R) by a sequence of 3-cycle reversals (in matrix form)  ∗ 1 0   ∗ 0 1   0 ∗ 1  −→  1 ∗ 0  , 1 0 ∗ 0 1 ∗ and edge-loop reversals (in case the 3-cycle reversal in T (R∗) uses the new row/ column) (again in matrix form):

i j i j i 0 1 −→ i 1 0 , j 0 1 j 1 0 arising from 0 1 0 0 0 1 0 0 1 −→ 1 0 0 . 1 0 0 0 1 0 Edge-loop reversals reverse the direction of an edge from a non-loop vertex i to a loop-vertex j and move the loop from j to i.

1.3.4 Hankel Tournaments and Their Generation

The main diagonal of an n × n matrix A = [aij] is the set of positions {(i, i) | 1 ≤ i ≤ n}. The Hankel diagonal (or anti-diagonal) of A is the set of positions 1.3. Tournaments and Tournament Matrices 39

{(i, n + 1 − i) | 1 ≤ i ≤ n}, illustrated for n = 4 in the matrix       .  

h 0 The Hankel transpose of A is the matrix A = [aij] obtained from A by transposing 0 across the Hankel diagonal and thus for which aij = an+1−j,n+1−i for all i and j. A Hankel tournament is deﬁned to be a tournament T for which T h = T . Thus, a (0, 1)-matrix T = [tij] is a Hankel tournament if and only if tii = 0 for all i, and tn+1−j,n+1−i = tij = 1 − tji = 1 − tn+1−i,n+1−j for all i =6 j. Except when the entry is on either the main or Hankel diagonals, one entry determines three others. The set of all Hankel tournaments with score vector R is denoted by TH (R). Example 1.3.10. An example of a Hankel tournament is  0 0 0 1 1   1 0 0 0 1    T =  1 1 0 0 0  .    0 1 1 0 0  0 0 1 1 0 As in the example, a Hankel tournament is combinatorially skew-symmetric (0 ↔ 1) about the main diagonal and symmetric (0 ↔ 0 and 1 ↔ 1) about the Hankel diagonal. The entries on the Hankel diagonal of a Hankel tournament can be 0 or 1 but, by the combinatorial skew-symmetry of a tournament, there must be bn/2c 1’s on the Hankel diagonal. If n is odd, then the entry t(n+1)/2,(n+1)/2 on the Hankel diagonal equals 0, since T is a tournament. Row i of a Hankel tournament T determines column i (tournament property), and determines row and column n + 1 − i (Hankel property). Example 1.3.11. Let T be an 8 × 8 Hankel tournament with row 3 given. Then, the entries in column 3, and row and column 8 + 1 − 3 = 6 are determined as shown in  0 1 − a g   0 1 − b f       a b 0 c d e f g     1 − c 0 d    ,  1 − d 0 c       1 − g 1 − f 1 − e 1 − d 1 − c 0 1 − b 1 − a     1 − f b 0  1 − g a 0 where the zeros on the main diagonal have been inserted, and the Hankel diagonal has been shaded. 40 Chapter 1. Some Combinatorially Deﬁned Matrix Classes

Let R = (r1, r2, . . . , rn) be the score vector of a Hankel tournament T , and let S = (s1, s2, . . . , sn) be the column sum vector of T . Since T is symmetric about the Hankel diagonal, ri = sn+1−i. Since T is a tournament, rn+1−i = (n − 1) − sn+1−i = (n − 1) − ri, and thus the score vector R satisﬁes the Hankel property that ri + rn+1−i = n − 1 for all i = 1, 2, . . . , n. In particular, if n is odd, r(n+1)/2 = (n − 1)/2. It can be easily veriﬁed, by using permutations preserving the tournament and Hankel properties, that there is no loss of generality in assuming that R is nondecreasing, and we now make this assumption.

Theorem 1.3.12. Let R = (r1, r2, . . . , rn) be a vector of nonnegative integers with r1 ≤ r2 ≤ · · · ≤ rn. Then, there exists a Hankel tournament with score vector Pk k R if and only if ri + rn+1−i = n − 1 for i = 1, 2, . . . , n, and i=1 ri ≥ 2 for k = 1, 2, . . . , n, and with equality if k = n. Thus, besides Landau’s conditions for the existence of a tournament with score vector R, the only other condition needed is that the score vector satisﬁes the Hankel property. (In the theorem it suﬃces to assume only that R is 2-nearly nondecreasing.) The proof of Theorem 1.3.12 proceeds by choosing a minimal counterexample (n minimum and then r1 minimum), which is one of the methods that have been used to prove Landau’s theorem. An algorithm to construct a tournament in TH (R) is given in Brualdi–Fritscher [8]. As with the classes T (R) and T `(R), all the tournaments in a nonempty class TH (R) can be generated from a given one by simple transformations with all intermediate tournaments in TH (R). The three transformations needed are: (i) 3-cycle Hankel reversals: reversing a 3-cycle on indices i,(n + 1)/2, n + 1 − i (when n is odd); (ii) two 3-cycle reversals consisting of a pure 3-cycle i → j → k → i and its complementary 3-cycle n + 1 − i → n + 1 − k → n + 1 − j → n + 1 − i; here pure means that {i, j, k} ∩ {n + 1 − i, n + 1 − j, n + 1 − k} = ∅; (iii) Hankel 4-cycle reversals: i → j → n + 1 − j → n + 1 − i → i.

1.3.5 Combinatorially Skew-Hankel Tournaments and Their Generation A combinatorially skew-Hankel tournament is an n×n (0, 1)-matrix which is combinatorially skew-symmetric about both the main diagonal and the Hankel diagonal, and which has only zeros on both its main diagonal and its Hankel diagonal. Let Dn be the n × n (0, 1)-matrix with ones on the main diagonal and on the Hankel diagonal, and zeros elsewhere. The n×n (0, 1)-matrix T = [tij] is a combinatorially t h skew-Hankel tournament if and only if T = Jn − Dn − T and T = Jn − Dn − T ; so T t = T h or, put another way, T th = T . Thus, T is a combinatorially skew- Hankel tournament if and only if tii = ti,n+1−i = 0 for all i, and tji = 1 − tij and 1.3. Tournaments and Tournament Matrices 41

tn+1−j,n+1−i = 1 − tij for all j =6 i, n + 1 − i. If T is a combinatorially skew-Hankel tournament, then tij = tn+1−i,n+1−j for all i and j.

Example 1.3.13. An example of a combinatorially skew-Hankel tournament is

 0 1 0 0 0   0 0 1 0 1     1 0 0 0 1  ,    1 0 1 0 0  0 0 0 1 0 where the four shaded entries determine all the other entries oﬀ the main and Hankel diagonals. A combinatorially skew-Hankel tournament is invariant under a rotation by 180 degrees.

Notice that a combinatorially skew-Hankel tournament is not a proper tournament because symmetrically opposite entries on the Hankel diagonal are both zero. If we view T with respect to either its main diagonal or its Hankel diagonal, we have a round-robin tournament in which, for each i, players i and n + 1 − i do not play a game. The set of all combinatorially skew-Hankel tournaments with a ∗ prescribed score vector R is denoted by TH (R). Let T = [tij] be an n × n combinatorially skew-Hankel tournament, and let the score vector of T be R = (r1, r2, . . . , rn). Since T is invariant under a rotation by 180 degrees, we have that, for each i, row n+1−i is obtained by reversing row i, and thus the score vector of T is palindromic, that is, R = (r1, r2, r3, . . . , r3, r2, r1).

Example 1.3.14. A 7 × 7 combinatorially skew-Hankel tournament with palindromic score vector R = (1, 3, 4, 2, 4, 3, 1) is given by

 0 0 1 0 0 0 0   1 0 0 1 0 0 1     0 1 0 1 0 1 1     1 0 0 0 0 0 1  .    1 1 0 1 0 1 0     1 0 0 1 0 0 1  0 0 0 0 1 0 0

As illustrated in the next example for n = 8, since T is combinatorially skew with respect to the main diagonal, row i of T not only determines row n + 1 − i, it also determines columns i and n + 1 − i.

Example 1.3.15. Let T be an 8 × 8 combinatorially skew-Hankel tournament with row 3 given. Then, the entries in column 3 and row and column 8 + 1 − 3 = 6 are 42 Chapter 1. Some Combinatorially Defined Matrix Classes determined as shown in  0 1 − a 1 − f 0   0 1 − b 1 − e 0       a b 0 c d 0 e f     1 − c 0 0 1 − d    ,  1 − d 0 0 1 − c       f e 0 d c 0 b a     0 1 − e 1 − b 0  0 1 − f 1 − a 0 where the zeros on the main and Hankel diagonals have been inserted and shaded. ∗ The determination of the nonemptiness of a class TH (R) depends on the parity of n. The score vector of a combinatorially skew-Hankel tournament is palindromic but, as is easily verified, there is no loss of generality in assuming that the first half of R is nondecreasing. Theorem 1.3.16. Let n be an even integer, and consider a vector of nonnegative integers R = (r1, r2, . . . , rn/2, rn/2, . . . , r2, r1) such that (r1, r2, . . . , rn/2) is nondecreasing. Then, there exists a combinatorially skew-Hankel tournament with score Pk vector R if and only if i=1 ri ≥ k(k − 1), for k = 1, 2, . . . , n/2, and with equality for k = n/2. In Theorem 1.3.16 it is enough to assume that R is 3-nearly nondecreasing. To verify the necessity of the conditions, consider  

 X1 X2      ,    

where X1 is k × k (k ≤ n/2) with zeros on its main diagonal, and X2 is k × k with zeros on its Hankel diagonal. Both X1 and X2 are k × k tournament matrices. k Thus, the sum of the entries of X1 and the sum of the entries of X2 are each 2 , Pk and together the sum is k(k − 1). Thus, i=1 ri ≥ k(k − 1). Theorem 1.3.17. Let n be an odd integer, and consider a vector of nonnegative integers R = (r1, . . . , r(n−1)/2, r(n+1)/2, r(n−1)/2, . . . , r1) such that (r1, r2, . . . , r(n−1)/2) is nondecreasing. Then, there exists a combinatorially skew-Hankel tournament with score vector R if and only if r(n+1)/2 ≤ n − 1 and

k + X r(n+1)/2 ri ≥ k(k − 1) + k − 2 i=1 for k = 1, 2,..., (n − 1)/2, and with equality if k = (n − 1)/2. 1.3. Tournaments and Tournament Matrices 43

An algorithm is given in Brualdi–Fritscher [8] to construct a combinatorially skew-Hankel tournament with score vector R. ∗ All tournaments in a nonempty class TH (R) can be generated from any one of them using two kinds of transformations with all intermediate tournaments in ∗ TH (R). The transformations are: (i) pairs consisting of a pure 3-cycle reversal of i → j → k → i and the complementary 3-cycle reversal of n + 1 − i, n + 1 − k → n + 1 − k → n + 1 − i (here again pure means that {i, j, k} ∩ {n + 1 − i, n + 1 − j, n + 1 − k} = ∅); (ii) 4-cycle skew-Hankel reversals, i → j → n + 1 − i → n + 1 − j → i. We conclude with the following observations. One could consider Hankel and combinatorially skew-Hankel loopy tournaments but nothing is gained since, in both cases, it can be shown that the score vector R determines which diagonal elements are 1 and which are 0. Also, by reversing the order of columns, combinatorially skew-Hankel H-loopy (i.e., with ones allowed on the Hankel diagonal) tournaments are equivalent to combinatorially skew-Hankel loopy tournaments. We may also consider combinatorially skew-Hankel doubly-loopy tournaments, that is, combinatorially skew-Hankel tournaments T = [tij] with possible ones on both the main diagonal and the Hankel diagonal. But, again, it can be shown that there is nothing essentially new here; see Brualdi–Fritscher [8]. Bibliography

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[14] R.A. Brualdi and M.W. Schroeder, “Alternating sign matrices and their Bruhat order”, Discrete Mathematics 340(8) (2017), 1996–2019. [15] M. Catral, M. Lin, D.D. Olesky, and P. van den Driessche, “Inverses and eigenvalues of diamond alternating sign matrices”, Spec. Matrices 2 (2014), 78–88. [16] A.B. Cruse, “On linear programming duality and Landau’s characterization of tournament scores”, Acta Univ. Sapientiae Informatica 6 (2014), 21–32. [17] R.C. Entringer and L.K. Tolman, “Characterizations of graphs having orien- tations satisfying local degree restrictions”, Czech. Math. J. 28 (1978), 108– 119. [18] M. Fortin, “The MacNeille completion of the poset of partial injective functions”, Electron. J. Combinatorics 15 (2008), #R62. [19] S.L. Hakimi, “On the degrees of the vertices of a directed graph”, J. Franklin Institute 279 (1965), 290–308. [20] S. Kirkland, private communication. [21] G. Kuperberg, “Another proof of the alternating sign matrix conjecture”, International Mathematics Research Notes (1966), 139–150. [22] H.G. Landau, “On dominance relations and the structure of animal societies. III. The condition for a score structure”, Bull. Math. Biophys. 15 (1953), 143–148. [23] A. Lascoux and M.P. Sch˝utzenberger, “Treillis et bases des groupes de Cox- eter”, Electron. J. Combin. 3 (1996), #R27. [24] H. MacNeille, “Partially ordered sets”, Trans. Amer. Math. Soc. 42 (1937), 416–460. [25] P. Magyar, “Bruhat order for two ﬂags and a line”, J. Algebraic Combinatorics 21 (2005), 71–101. [26] W.H. Mills, D.P. Robbins, and H. Rumsey, “Alternating sign matrices and descending plane partitions”, J. Combin. Theory, Ser. A 34 (1983), 340–359. [27] E. Tyrtyshnikov, “Matrix Bruhat decompositions with a remark on the QR (GR) algorithm”, Linear Algebra Appl. 250 (1997), 61–68. [28] D. Zeilberger, “Proof of the alternating sign matrix conjecture”, Electron. J. Combinatorics 3 (1966), #R13. Chapter 2

Sign Pattern Matrices by P. van den Driessche

2.1 Introduction to Sign Pattern Matrices

The study of sign pattern matrices is an important part of combinatorial matrix theory. It has a rich theory in its own right and, in addition, some results are useful in applications to dynamical systems where sign patterns arise naturally, for example in predator-prey populations, economics, chemistry, and sociology. The aim of this chapter is to describe some spectral properties of matrices with a given sign pattern from the perspective of combinatorial matrix theory, showing some results, techniques, applications, and open problems. Topics are chosen from the literature on sign patterns; some of these and other related topics can be found in the book Brualdi–Shader [7], and in the chapter Hall–Li [34] from the Handbook of Linear Algebra edited by L. Hogben, and references therein. Readers are encouraged to consult these other references for topics beyond those in this chapter, which is a somewhat personal overview of some results on sign pattern matrices. Techniques employed involve analysis, combinatorics, directed graphs (digraphs), matrix theory, and linear algebra, and numerical examples are given. The next subsection gives some basic deﬁnitions used throughout the chapter, before beginning the main sections on potential stability, spectrally arbitrary sign patterns, reﬁned inertia of sign patterns, and inertially arbitrary patterns.

2.1.1 Notation and Deﬁnitions

An n × n sign pattern (matrix) S = [sij] is a matrix with sij ∈ {+, −, 0}. Note that in Brualdi–Shader [7], + and − are replaced with +1 and −1, respectively. 48 Chapter 2. Sign Pattern Matrices

Associated with a sign pattern is a sign pattern class of real square matrices, n×n namely Q(S) = {A = [aij] ∈ R | sign(aij) = sij for all i, j}. If matrix A ∈ Q(S), then A is a (matrix) realization of S, and S = sgn(A) is the sign pattern of A. Here, all sign patterns are square, and all matrices have real entries. For a given property P , a sign pattern S requires P if every matrix A ∈ Q(S) has property P , and allows P if there exists at least one matrix A ∈ Q(S) that has property P . Note that S requires P implies that S allows P . − + −2 0.1 Example 2.1.1. If S = [ + − ], then A = [ π −7 ] ∈ Q(S) is a matrix realization of S. Sign pattern S requires a negative trace and allows (but does not require) a positive determinant. The signed digraph D(S) of an n × n sign pattern S has vertex set {1, . . . , n} and a positive (negative) arc from i to j if and only if sij is positive (negative). A (simple, directed) cycle of length k (a k-cycle) is a sequence of k arcs (i1, i2),..., (ik, i1) such that the vertices i1, . . . , ik are distinct. A cycle is positive (negative) if there is an even (odd) number of negative arcs on the cycle.

2.2 Potential Stability of Sign Patterns 2.2.1 Stability Definitions In keeping with the convention of dynamical systems (rather than that usually used in matrix theory), the following definition of stability is used here. Matrix A is (negative) stable if each of its eigenvalues has a negative real part. This leads to two important concepts of stability for sign patterns. A sign pattern S is potentially stable if there exists a stable matrix A ∈ Q(S), that is, if S allows stability. A sign pattern S is sign stable if every A ∈ Q(S) is a stable matrix, that is, if S requires stability. Conditions for potential or sign stability of S are often given in terms of the signed digraph D(S) of S, as is the following definition. A sign pattern S is a tree sign pattern if D(S) is strongly connected and has no k-cycles for k ≥ 3, i.e., has only 2-cycles and 1-cycles, the latter are also called loops. Examples are path sign patterns and star sign patterns, where an n × n star sign pattern has a center vertex with degree n − 1 connected to n − 1 leaf vertices. Example 2.2.1. Sign pattern

− + 0  S = + − + 0 − 0 is a potentially stable tree (path, star) sign pattern, since the matrix

−1 1 0 A =  2 −2 1 ∈ Q(S) 0 −2 0 2.2. Potential Stability of Sign Patterns 49 has eigenvalues approximately −2.52, −0.240.86i. However, S is not sign stable, since changing the (2,2) entry of A to −1 gives a matrix with eigenvalues −2, i. Changing the (1, 1) entry of S to + gives a sign pattern that requires a positive determinant, so it is not even potentially stable.

In 1947, Samuelson [49] considered qualitative problems in economics involving sign patterns, and this is usually regarded as the first work on sign patterns. About twenty years later, Maybee–Quirk [45] studied these from a matrix and digraph point of view and wrote: “Specification of necessary and sufficient conditions for potential stability remains an unsolved problem”. Apart from a few special cases, this remains true today. By contrast, sign stability was characterized by Jeffries–Klee–van-den-Driessche [38], and an algorithm was given to test whether or not a sign pattern is sign stable. Since the 1970s researchers have derived many results about sign patterns and applied some of them to dynamical systems, e.g., in economics and food webs.

2.2.2 Stability of a Dynamical System

Assume a dynamical system modeled by a set of n ordinary diﬀerential equations is at an equilibrium x∗ ∈ Rn. Considering small perturbations and linearizing about x∗, the time evolution is governed by

dx(t) = Ax(t) dt

At for some n × n community matrix A. Solutions are of the form x(t) = e x0 and if A is a stable matrix, then perturbations die out and x∗ is a locally asymptotically stable equilibrium. On the other hand, if A has an eigenvalue with a positive real part, then perturbations grow and x∗ is unstable. However, in applications, it is often the case that the magnitudes of entries in A are unknown whereas the signs are known (for example in a grass-rabbit-fox system: rabbits eat grass, foxes eat rabbits) so A should be regarded as a sign pattern S with digraph D(S) given in Fig. 2.1. The community matrix for this grass-rabbit-fox system about an equilibrium with positive amounts of grass, rabbits, and foxes has the sign pattern

 0 − 0  S = + 0 − , 0 + − which is in fact sign stable (and thus potentially stable). Therefore, the equilibrium x∗ is locally asymptotically stable and, since S is sign stable, this is independent of the magnitudes of the system parameters, e.g., the rate at which rabbits eat grass. April 19,50 2010 Chapter 2. Sign Pattern Matrices

q − q −

1 1 j − + + wvutpqrs ~}|xyz{ ~}|xyz{

Figure 2.1: The signed digraph of the community matrix for a grass-rabbit-fox system.

2.2.3 Characterization of Sign Stability It is quite restrictive for a sign pattern to be sign stable, as the following characterization shows. The notation S[β | γ] denotes the subpattern of S in rows indexed by β and columns indexed by γ, and βc denotes the complement of β. A sign pattern S is sign nonsingular if the determinant of A is nonzero for all A ∈ Q(S), that is, S requires a nonzero determinant. The wording of the theorem characterizing sign stability mostly follows Brualdi–Shader [7, Thm. 10.2.2].

Theorem 2.2.2. Let S = [sij] be an n × n irreducible sign pattern. Then, S is sign stable if and only if

(i) sii ≤ 0 for i = 1, . . . , n (i.e., all loops in D(S) are negative);

(ii) sijsji ≤ 0 for i =6 j (i.e., all 2-cycles in D(S) are negative); (iii) D(S) is a tree sign pattern; (iv) S is sign nonsingular; and (v) there does not exist a nonempty subset β of {1, 2, . . . , n} such that each diagonal entry of S[β | β] is zero, each row of S[β | β] contains at least one nonzero entry, and no row of S[βc | β] contains exactly one nonzero entry. If S satisfies the first three conditions of the above theorem, then S is sign semi-stable, having all its eigenvalues in the nonpositive half plane. The last two conditions rule out zero eigenvalues and nonzero pure imaginary eigenvalues, respectively. An important example illustrating the last condition is provided by the following example from Jeffries [37]. Example 2.2.3.  0 + 0 0 0  − 0 + 0 0    S =  0 − − + 0     0 0 − 0 + 0 0 0 − 0 2.2. Potential Stability of Sign Patterns 51 is a tree (path) sign pattern and satisfies the first three conditions of Theorem 2.2.2 for sign stability, so it is sign semi-stable, i.e., all eigenvalues have nonpositive real parts. In addition (from the product s12s21s33s45s54), S requires a nonzero determinant. However, S fails the last condition: taking β = {1, 2, 4, 5}, each row of S[β] contains a nonzero entry, and S[βc | β] = S[3 | 1245] has two nonzero entries. Numerically, if A ∈ Q(S) has the magnitude of each nonzero entry equal to 1, then A has eigenvalues i, with the other three eigenvalues having negative real parts.

2.2.4 Basic Facts for Potential Stability Potential stability is preserved under transposition, permutation similarity, and signature similarity (but not under negation). Two sign patterns are equivalent in this context if one is obtained from the other by any combination of these three operations. Thus lists of potentially stable sign patterns are usually given only up to equivalence. A pattern Sb is a superpattern of S (and S is a subpattern of Sb) if sbij = sij whenever sij =6 0, and a pattern is a superpattern (and subpattern) of itself. If S is potentially stable, then by continuity (since the eigenvalues of a matrix depend continuously upon its entries) any superpattern is also potentially stable. The direct sum S1 ⊕ S2 is potentially stable if and only if S1 and S2 are potentially stable. Thus, only irreducible potentially stable sign patterns need to be considered. A sign pattern S is minimally potentially stable if it is irreducible and potentially stable, and replacing any + or − by 0 (i.e., taking any proper subpattern) gives a sign pattern that is not potentially stable. Note that the subpattern may be reducible. In view of the above facts, minimally potentially stable patterns are especially important. If an n × n sign pattern S is potentially stable, then there exists A ∈ Q(S) such that the sum of the k × k principal minors is (−1)k for k = 1, 2, . . . , n. This result comes from considering the characteristic polynomial of a stable matrix A. In particular at least one aii must be negative, and the sign of the determinant of A must be (−1)n. A potentially stable sign pattern S must have a negative diagonal entry, and any S with all diagonal entries negative is potentially stable. However, positive feedback may promote stability, as the following somewhat surprising 4×4 example shows. Example 2.2.4 (Bone, [5]).

− + + 0  − s22 0 0  S =   + 0 0 + 0 + 0 0 is potentially stable if s22 is + but not potentially stable if s22 is − or 0. These 52 Chapter 2. Sign Pattern Matrices statements are proved by using the Routh–Hurwitz conditions; see, e.g., Horn– Johnson [36, Sect. 2.3].

2.2.5 Known Results on Potential Stability for Small Orders Johnson–Summers [40] compiled a list of potentially stable tree sign patterns of orders n = 2, 3, 4. For n = 4, Johnson–Maybee–Olesky–van-den-Driessche [39] gave one more and Pang found three more minimally potentially stable path sign patterns; see Lin–Olesky–van-den-Driessche [43]. Miyamichi [46] lists all minimally potentially stable sign patterns of order 3 (and gives some constructions for general n). The minimally potentially stable sign patterns (up to equivalence) are now given for n = 2 and 3, using a sign pattern for n = 2 but digraph representations for n = 3. There is one minimally potentially stable sign pattern of order 2, namely − + [ − 0 ], which is sign stable. There are ﬁve minimally potentially stable sign patterns of order 3, two of which are tree (path) sign patterns, with one being sign stable, while the remaining three each contain a 3-cycle. These ﬁve are given in Fig. 2.2.

2.2.6 Suﬃcient Condition for Potential Stability Let det B[{1, . . . , k}] denote the minor of B on rows and columns 1, . . . , k. A sign pattern S allows a nested sequence of properly signed principal minors if there exist a matrix A ∈ Q(S) and a permutation matrix P such that B = P AP T has sign(det B[{1, . . . , k}]) = (−1)k, for k = 1, . . . , n. The following result relating such a nested sequence to potential stability is from Johnson–Maybee–Olesky–van-den-Driessche [39, Thm. 2.1]. Theorem 2.2.5. If S is an n×n sign pattern allowing a nested sequence of properly signed principal minors, then S is potentially stable. The proof of this theorem uses a result from Fisher–Fuller [24] and Ballan- tine [2], which shows that if an n × n matrix A has a nested sequence of properly signed principal minors, then there exists a positive diagonal matrix D such that DA is stable. Note that the sign pattern of DA is the same as that of A. The converse of the above theorem is false for n = 3 and even false for tree sign patterns with n = 4. As an example, for n = 3 the potentially stable sign pattern

− + 0   0 0 + + − 0 does not have a nested sequence of properly signed principal minors, but is minimally potentially stable. Its signed digraph is the third one with a 3-cycle given in Fig. 2.2. 2.2. Potential Stability of Sign Patterns 53

− Ù t − t − 1 4 2 4 3 + +

+ − Ù Ù t − t + 1 4 2 4 3 + +

− − Ù Ù + / 1O 2

− Ð + 3

− − Ù Ù t − 4 + / 1O 2 1O F 2 + + − + + Ð Ö − 3 3

Figure 2.2: Digraphs of all (up to equivalence) minimally potentially stable sign patterns for n = 3. 54 Chapter 2. Sign Pattern Matrices

Care must be taken with magnitudes when ﬁnding nested sequences, as illustrated by the following example from Johnson et al. [39, Ex. 1.1]. Example 2.2.6. Consider the sign pattern − + 0  S = − + + . 0 + +

For A ∈ Q(S), the conditions det A[1] < 0, det A[12] > 0, and det A < 0 are not simultaneously realizable, even though each one is separately realizable. In fact, S is not potentially stable. For a restricted class of tree sign patterns, a nested sequence of properly signed principal minors is a necessary and suﬃcient condition for potential stability; see Johnson et al. [39, Thm. 4.2].

Theorem 2.2.7. Let S be a tree sign pattern with exactly one nonzero sii (which is negative). Then, S is potentially stable if and only if S allows a nested sequence of properly signed principal minors. For star sign patterns, Gao–Li [26] in their Theorems 3.5 and 4.2 characterized all n × n potentially stable patterns in terms of the number of positive, negative, and zero diagonal entries and the signs of 2-cycles. They note that for a star sign pattern to allow a nonzero determinant (and thus have a chance of being potentially stable), at most one of the leaf vertices may not have a loop.

2.2.7 Construction of Higher Order Potentially Stable Sign Patterns In this section, two methods are given to construct a higher-order irreducible potentially stable sign pattern starting with a known small-order sign pattern that is potentially stable. Some other similar constructions are given in Grundy– Olesky–van-den-Driessche [33]. A construction involving cycles is given by Miyamichi [46]. Start with the potentially stable sign pattern given by the three-vertex digraph as shown in Fig. 2.3 (the ﬁrst tree sign pattern listed in Fig. 2.2). The second digraph in Fig. 2.3 on at least ﬁve vertices results from relabelling vertex 2 as n − 1, and vertex 3 as n; replacing the 2-cycle on vertices 1 and 2 by an (n − 1)-cycle; and putting negative loops on vertices 2, . . . , n − 2. This resulting sign pattern is potentially stable. Here the arc from n − 1 to 1 can be positive or negative (but not zero, as that would yield a pair of pure imaginary eigenvalues). The proof is divided into two cases depending on the sign of this arc, which in turn determines the sign of the (n − 1)-cycle. If the (n−1)-cycle is negative, then the sign pattern has a nested sequence of properly signed principal minors, implying by Theorem 2.2.5 that it is potentially stable. 2.2. Potential Stability of Sign Patterns 55

− Ù t − t − 1 4 2 4 3 + +

y r − / / / n − 1 4 n 1 U + 2 U + 3 U + − − −

Figure 2.3: Construction of an order n potentially stable sign pattern from one of order 3.

If the (n − 1)-cycle is positive, then the characteristic polynomial of a matrix 2 A in the sign pattern class is (z − C2)(z − a11) ··· (z − an−2,n−2) − Cn−1z, where C2 denotes the product of entries of A on the 2-cycle, Cn−1 denotes the product of entries of A on the (n − 1)-cycle, and aii are the diagonal entries resulting from the loops on vertices 1,... ,n − 2. Gantmacher [25, Thm, 13, p. 228] states that a polynomial f(z) = h(z2) + zg(z2) with positive coeﬃcients is stable if and only if the zeros α1, α2,... of h(u) and the zeros β1, β2,... of g(u) are all negative and satisfy the interlacing property 0 > α1 > β1 > α2 > β2 > ··· . Here, the magnitudes of C2,Cn−1 can be chosen to satisfy these conditions, and thus obtain stability of the order n irreducible sign pattern with signed digraph as the second one depicted in Fig. 2.3. In Kim–Olesky–Shader–van-den-Driessche–van-der-Holst–Vander-Meulen [41] the authors give a method to take a stable matrix and produce a stable matrix (and thus a potentially stable sign pattern) of higher order. A special case of this is now stated. Theorem 2.2.8. Let A be an n × n stable matrix, and let u and x be n-vectors so that xT u = k, a positive scalar. Then, the (n + 1) × (n + 1) matrix B given by I 0 A u I 0 A − uxT u B = n n = xT 1 0 −k −xT 1 xT A 0 is stable with spectrum σ(B) = σ(A) ∪ {−k}. This construction is illustrated with the following companion matrix. Example 2.2.9.  0 1 0  A =  0 0 1  −1 −3 −3 is a stable matrix (having -1 as an eigenvalue with algebraic multiplicity three) giving a minimally potentially stable sign pattern, with signed digraph equivalent 56 Chapter 2. Sign Pattern Matrices to the second one with a 3-cycle given in Fig. 2.2. Applying the theorem with u = [0, 0, 1]T , x = [3, −3, 1]T , and xT u = k = 1 gives

 0 1 0 0   0 0 1 0  B =   ,  −4 0 −4 1  −1 0 −6 0 with σ(B) = σ(A) ∪ {−k}. Thus, B has eigenvalue -1 with algebraic multiplicity four, and gives a (minimally) potentially stable sign pattern of order 4 (with no nested sequence of properly signed principal minors).

2.2.8 Number of Nonzero Entries The minimum number of entries in an irreducible potentially stable sign pattern has been recently investigated by Grundy–Olesky–van-den-Driessche [33], who proved that an n × n irreducible sign pattern with a properly signed nest (so, it is potentially stable) has at least 2n − 1 nonzero entries. But there are examples of potentially stable sign patterns with fewer than this number of nonzero entries, as the next example with n = 4 illustrates. Example 2.2.10. Consider the stable matrix

 0 1 0 1   0 0 1 0  A =   ,  −1 0 0 0  −3 0 0 −1 which gives a potentially stable sign pattern with 6 = 2n−2 nonzero entries. Note that this pattern does not allow a nested sequence of properly signed principal minors. For small orders, Grundy–Olesky–van-den-Driessche [33] found the following results: (i) an n = 2 irreducible potentially stable sign pattern allows a properly signed nest, so it has at least 2n − 1 = 3 nonzero entries; (ii) an n = 3 irreducible potentially stable sign pattern has at least 2n − 1 = 5 nonzero entries (from the list in section 2.2.5); (iii) an n = 4 or 5 irreducible potentially stable sign pattern has at least 2n − 2 (i.e., 6 or 8) nonzero entries; (iv) an n = 6 irreducible potentially stable sign pattern has at least 2n − 3 = 9 nonzero entries; (v) there is an n = 7 (n = 8, n = 9) irreducible potentially stable sign pattern with 2n − 3 = 11 (2n − 3 = 13, 2n − 4 = 14) nonzero entries. 2.3. Spectrally Arbitrary Sign Patterns 57

The results for n = 7, 8, 9 come from examples, but it is not known whether there exist any with fewer than these numbers of nonzero entries. The proof for n = 4 starts by noting that a potentially stable sign pattern S has at least one negative diagonal entry. Since S is irreducible it must have at least four nonzero oﬀ-diagonal entries. If S has at least ﬁve, then the result holds (giving six nonzero entries). Otherwise, if S has only four, then D(S) contains a cycle of length 4. But in order for S to have a positive principal minor of order 2 (as required for potential stability, see section 2.2.4) there must be at least one more nonzero entry, giving 1+ 4 + 1 = 6 arcs, that is six nonzero entries in S.

2.2.9 Open Problems Related to Potential Stability To conclude this section on potential stability, here are some open problems. (i) Give necessary and sufficient conditions for a sign pattern to be minimally potentially stable. Currently this is open and there are no complete lists of such sign patterns for tree sign patterns of order 5, or for general sign patterns of order 4. (ii) Give easily verifiable conditions to show that a sign pattern is not potentially stable. This would be useful in the context of the stability of solutions of differential equations derived from dynamical systems in applications. (iii) Find an easy way to determine if a sign pattern allows or requires a properly signed nest. (iv) Find the minimum number of nonzero entries in an n × n irreducible potentially stable sign pattern for n ≥ 7.

2.3 Spectrally Arbitrary Sign Patterns 2.3.1 Some Deﬁnitions Relating to Spectra of Sign Patterns The inertia of a real n × n matrix A is the triple of nonnegative integers i(A) = (n+(A), n−(A), n0(A)), in which n+(A), n−(A), n0(A) are the number of eigenvalues of A with positive, negative, and zero real parts (counting multiplicities), respectively. So, n+(A) + n−(A) + n0(A) = n. The inertia of a sign pattern S is i(S) = {i(A) | A ∈ Q(S)}. − + Example 2.3.1. S = [ − 0 ] is a sign stable pattern, thus, i(S) = {(0, 2, 0)}, a unique inertia. − + Example 2.3.2. If S = [ + − ], then i(S) = {(0, 2, 0), (0, 1, 1), (1, 1, 0)}. This is found −a b by taking A = [ c −d ] ∈ Q(S) with a, b, c, d > 0. The characteristic polynomial of A is z2 + (a + d)z + ad − bc, giving i(A) = (0, 2, 0) if ad > bc, i(A) = (0, 1, 1) if ad = bc, and i(A) = (1, 1, 0) if ad < bc. 58 Chapter 2. Sign Pattern Matrices

An n × n inertially arbitrary pattern (IAP) S has (n+, n−, n0) ∈ i(S) for all nonnegative integers satisfying n+ + n− + n0 = n. Equivalently, that S is an IAP means that S allows all (n + 1)(n + 2)/2 possible inertias. For example, if n = 2 there are 6, and if n = 3 there are 10 possible inertias. A sign pattern S is a minimal IAP if no proper subpattern of S is an IAP. To be more precise than inertia, the spectrum of a sign pattern is considered. An n × n sign pattern S is a spectrally arbitrary pattern (SAP) if for each real monic polynomial r(z) with degree n, there exists A ∈ Q(S) with characteristic polynomial equal to r(z), that is, S allows all possible spectra of a real matrix. The restriction to a real matrix imposes the condition that if α + iβ with β =6 0 is an eigenvalue, then α − iβ is also an eigenvalue, i.e., if S is a SAP, then S has any self-conjugate multiset of n complex numbers as the spectrum of some A ∈ Q(S). A sign pattern S is a minimal SAP if no proper subpattern of S is a SAP. It follows from the definitions that if S is a SAP, then S is an IAP. However, the converse is false for n ≥ 4: for example, Cavers–Vander-Meulen [13] give a 4 × 4 minimal IAP sign pattern that does not allow four nonzero pure imaginary eigenvalues. To investigate SAPs, one more definition is important. A matrix A is nilpotent of index q if Aq = 0 and q is the smallest such positive integer. A pattern S is potentially nilpotent (PN) if there exists A ∈ Q(S) such that A is nilpotent, i.e., Aq = 0 for some q > 0, or A has all zero eigenvalues. From the definitions, if S is a SAP, then S is potentially nilpotent, but it is not in general true that an IAP is potentially nilpotent. The smallest example of an IAP not PN is the 5 × 5 sign pattern G5 given in Kim–Olesky–van-den-Driessche [42]; see also Cavers–Garnett– Kim–Olesky–van-den-Driessche [11]. Inertially arbitrary, spectrally arbitrary, and potentially nilpotent sign patterns are preserved under negation, transposition, permutation similarity, and signature similarity. Two sign patterns are equivalent in this context if one can be obtained from the other by any combination of these four operations. Thus lists of such sign patterns are generally given only up to equivalence. To illustrate the above definitions, here are two sign patterns of order 2.

− + Example 2.3.3. S = [ + − ] is not an IAP or SAP since (2, 0, 0) ∈/ i(S). Additionally S is not PN as S requires a negative trace.

− + Example 2.3.4. Let T2 = [ − + ]. It can be shown directly that any monic quadratic polynomial can be achieved as the characteristic polynomial of a matrix in Q(T2). The pattern T2 is a minimal SAP and, up to equivalence, T2 is the unique 2 × 2 SAP. It is also PN and the unique minimal 2 × 2 IAP.

Since a SAP must allow any characteristic polynomial, an obvious necessary condition for a SAP is given in terms of principal minors.

Theorem 2.3.5. If S is an n × n SAP, then S allows a positive and a negative principal minor of each order k for 1 ≤ k ≤ n. 2.3. Spectrally Arbitrary Sign Patterns 59

2.3.2 A Family of Spectrally Arbitrary Sign Patterns Drew–Johnson–Olesky–van-den-Driessche [20] introduced the deﬁnition of a SAP and considered the family of tree (path) sign patterns − + 0  − 0 +     .. ..  Tn =  − . .     .   .. 0 + 0 − + for n ≥ 2. They developed the Nilpotent-Jacobian method for proving that a sign pattern is a SAP, used it to show that Tn is a minimal SAP for small values of n, and conjectured that this is true for all n. This Nilpotent-Jacobian method is now stated; its proof uses the Implicit Function Theorem. From the statement, it becomes clear that potentially nilpotent matrices play a large role in utilizing this method. Theorem 2.3.6 (Nilpotent-Jacobian Method). Let S be an n × n sign pattern, and suppose that there exists some nilpotent matrix A ∈ Q(S) with at least n nonzero entries, say ai1j1 , . . . , ainjn . Let X be the real matrix obtained by replacing these entries in A by variables x1, . . . , xn, and let the characteristic polynomial of X be n n−1 n−2 n−1 n given by p(z) = z − p1z + p2z − · · · + (−1) pn−1z + (−1) pn, where pi = pi(x1, . . . , xn) is diﬀerentiable in each xj. If the n × n Jacobian matrix with

(i, j) entry equal to ∂pi/∂xj is nonsingular at (x1, . . . , xn) = (ai1j1 , . . . , ainjn ), then every superpattern of S (including S itself) is spectrally arbitrary. Example 2.3.7. Use of the Nilpotent-Jacobian method is now illustrated for  − + 0  T3 =  − 0 +  , 0 − +

 −1 1 0  which is PN, since A =  −1/2 0 1  ∈ Q(T3) is nilpotent. Letting 0 −1/2 1

 −1 1 0  X =  −x1 0 1  , 0 −x2 x3

3 2 with xi > 0, the characteristic polynomial of X is z + (1 − x3)z + (x1 + x2 − x3)z + x2 − x1x3. This gives the Jacobian matrix from Theorem 2.3.6 as  0 0 −1  J =  1 1 −1  , −x3 1 −x1 60 Chapter 2. Sign Pattern Matrices

which has determinant −1(1 + x3) = −2 at x3 = 1, so J is nonsingular. Thus, by the Nilpotent-Jacobian method, T3 is a SAP, and all its superpatterns are also SAP’s. If the (3,3) entry of T3 is set to zero, then the pattern is sign stable, see Theorem 2.2.2, whereas if the (1,1) entry is set to zero, then the resulting sign pattern has fixed inertia {(3, 0, 0)}. If an off-diagonal entry of T3 is set to zero, then the sign pattern requires either a positive or a negative eigenvalue. It follows that T3 is a minimal SAP. Drew–Johnson–Olesky–van-den-Driessche [20] used the Nilpotent-Jacobian method to prove that Tn is a minimal SAP for n = 2,..., 7, and Elsner–Olesky– van-den-Driessche [21] used this method with Maple to extend the result to n = 16. Then, in 2012, Garnett–Shader [31] proved that Tn is a minimal SAP for all n. They used the nilpotent matrices given in Behn–Driessel–Hentzel–Vander-Velden– Wilson [3]. A sketch of the proof of this important result, which confirmed the twelve year old conjecture, is given next; full details can be found in the cited references. Behn et al. [3] showed that, for n ≥ 2, each member of the tridiagonal family of matrices Fn ∈ Q(Tn) is nilpotent, where

 −f1 f1 0   −f2 0 f2     . .   .. ..  Fn =  −f3     ..   . 0 fn−1  0 −fn fn with fk = 0.5 csc((2k − 1)π/2n). They proved this using Chebyshev polynomials. Garnett–Shader [31] noted that Fn is similar to a matrix in a special form. Since Fn is a tridiagonal matrix, for j = 1, . . . , n − 1, the (j, j + 1) and (j + 1, j) entries always appear as a product in its characteristic polynomial and so, can be p p replaced by fjfj+1 and - fjfj+1, respectively. This new matrix Fñ is diagonally similar to Fn and was used by Garnett and Shader as the nilpotent realization of Tn. In addition, since csc(θ) = csc(π − θ), the entries in Fñ satisfy f˜1 = −fñ and f˜j,j+1 = fñ−j,n+1−j. Using these observations, this special form is illustrated for n = 4 as √  −f1 f1f2 0 0  √ √  − f f 0 f f 0  ˜  1 2 2 3  F4 =  √ √  . 0 − f2f3 0 f1f2  √  0 0 − f1f2 f1

Garnett–Shader [31, 32] then showed that F˜n satisﬁes the Nilpotent-Jacobian method (Theorem 2.3.6) by using the centralizer of this matrix, where B is in the centralizer of A if AB = BA. They went on to generalize this method and developed the following theorem. 2.3. Spectrally Arbitrary Sign Patterns 61

Theorem 2.3.8 (Nilpotent-Centralizer Method, [32, Thm. 3.7]). Let S be an n × n sign pattern and A ∈ Q(S) be nilpotent of index n. If the only matrix B in the centralizer of A for which the Hadamard (entrywise) product B ◦ AT = 0 is the zero matrix, then every superpattern of S is a SAP. They used this as a way of establishing the nonsingularity of the Jacobian matrix, and proved that Tn and all its superpatterns are SAPs. In fact, Tn is a minimal SAP, as can be checked by considering all proper subpatterns of Tn.

2.3.3 Minimal Spectrally Arbitrary Patterns and Number of Nonzero Entries Chronologically the ﬁrst n × n families of minimal SAPs were given in Britz– McDonald–Olesky–van-den-Driessche [6]. For example, the Hessenberg sign pattern Vn for n ≥ 2 is a SAP where

 + −   + − 0     . .   . ..  Vn =   .  + −     + 0 −  + −

They showed explicitly that the characteristic polynomial of Vn can be any arbitrary monic polynomial of degree n. Considering all possible cases for 3×3 sign patterns, Cavers–Vander–Meulen [13] proved the following result. Theorem 2.3.9. If S is a 3 × 3 sign pattern, then the following are equivalent: (i) S is a SAP; (ii) S is an IAP; (iii) up to equivalence, S is a superpattern of one of

− + 0  + − + + − 0  + + − − 0 + , + − 0  , + 0 − , + 0 − . 0 − + + 0 − + 0 − + 0 −

Note that the ﬁrst sign pattern is T3, and the second is also a tree sign pattern. Britz–McDonald–Olesky–van-den-Driessche [6] considered the minimum number of nonzero entries in a SAP, proved the following theorem, and stated a conjecture. Theorem 2.3.10. An n × n irreducible SAP has at least 2n − 1 nonzero entries. 62 Chapter 2. Sign Pattern Matrices

The proof begins by noting else than by using a positive diagonal similarity, n − 1 entries of the sign pattern can be set to 1. Then, at least n algebraically independent variables are needed to allow any monic characteristic polynomial of order n. Conjecture 2.3.11. For n ≥ 2 an n × n irreducible sign pattern that is spectrally arbitrary has at least 2n nonzero entries. This conjecture, which has become known as the 2n−conjecture, is true for n = 2 (T2), n = 3 (see Theorem 2.3.9), n = 4 (see Corpuz–McDonald [16]), and n = 5 (see DeAlba–Hentzel–Hogben–McDonald–Mikkelson–Pryporova–Shader– Vander-Meulen [19]), and for all tree sign patterns, but is still open for general sign patterns with n ≥ 6. Pereira [48] emphasized the relevance of potentially nilpotent sign patterns to the study of SAPs by considering full sign patterns, where a full sign pattern S has all sij nonzero. Theorem 2.3.12 (Pereira, [48, Thm. 1.2]). Any potentially nilpotent full sign pattern is a SAP. Pereira’s clever proof uses a perturbation of the Jordan normal form of a nilpotent n × n matrix N of index n with a given full sign pattern, combined with a companion matrix argument. The Nilpotent-Centralizer method (Theorem 2.3.8)) was used in Garnett– Shader [32] to give another proof that a full sign pattern is a SAP if and only if it is PN, since for a full sign pattern, the only matrix B that satisﬁes B ◦ N T = 0 is the zero matrix. Their method was also used to give necessary and suﬃcient conditions for other sign patterns to be spectrally arbitrary; for example a full tridiagonal sign pattern (i.e., no zero entry on the main, super- or sub-diagonals) is a SAP if and only if it is PN; see [32, Thm. 4.7].

2.3.4 Reducible Spectrally Arbitrary Sign Patterns While the results described above have concentrated on irreducible SAPs, here are a couple of results on reducible patterns. DeAlba et al. [19] noted that the direct sum of sign patterns of which at least two are of odd order is not a SAP (since the pattern requires at least two real eigenvalues). They also gave the following example of a direct sum of two sign patterns that is a SAP with only one of the summands being a SAP. Example 2.3.13. The sign pattern

 + + − 0   − − + 0  M4 =    0 0 0 −  + + 0 0 2.3. Spectrally Arbitrary Sign Patterns 63

is not a SAP, but M4 ⊕T2 is a SAP. They showed that M4 is not a SAP by ﬁnding exactly which real monic polynomials of degree 4 can be realized by a matrix in Q(M4) and showing that there are some such polynomials that cannot be realized. They then proceeded to show that M4 ⊕ T2 is a SAP by writing any given monic polynomial r(z) of degree 6 in quadratic and linear factors and ﬁnding a subset of the factors that can be realized as the characteristic polynomial of A1 ∈ Q(M4). Since T2 is a SAP, there is a matrix A2 ∈ Q(T2) having the product of the remaining factor(s) as its characteristic polynomial. Thus, A1 ⊕ A2 ∈ Q(M4 ⊕ T2) has r(z) as its characteristic polynomial. Note that the same method of proof shows that M4 ⊕ S is a SAP if S is any SAP.

2.3.5 Some Results on Potentially Nilpotent Sign Patterns From the above results on spectrally arbitrary sign patterns, the importance of nilpotent realizations is apparent. In fact, the study of potentially nilpotent sign patterns predates the study of SAPs and IAPs; see, for example, Eschenbach– Johnson [22]. Eschenbach–Li [23] characterized 3 × 3 patterns allowing nilpotence of index 2; Gao–Li–Shao [27] characterized 3 × 3 patterns allowing nilpotence of index 3 and gave constructions for n×n patterns allowing nilpotence of index 3. In addition, Catral–Olesky–van-den-Driessche [8] proved that the minimum number of nonzero entries in an irreducible n × n PN pattern is n + 1. For example,

 0 + 0 +   0 0 + 0  S =    + 0 0 0  0 0 − 0 is irreducible, has ﬁve nonzero entries, and allows nilpotence of index 4, since if A ∈ Q(S) has every entry of magnitude 1, then A4 = 0 whereas A3 =6 0. MacGillivray–Tiefenbach–van-den-Driessche [44] characterized PN star sign patterns, and proved that if S is a star sign pattern that is potentially nilpotent and potentially stable, then S is a SAP. By determining which of the 4 × 4 potentially stable path sign patterns (tridiagonal) are SAPs, Arav–Hall–Li–Kaphle– Manzagol [1] showed that if S is a 4 × 4 tree sign pattern that is potentially nilpotent and potentially stable, then S is a SAP. Kim et al. [41] gave the following construction for full PN sign patterns and thus, by Theorem 2.3.12, for SAPs. Theorem 2.3.14. Let A and U be n×n full sign patterns, with nilpotent A ∈ Q(A) and U ∈ Q(U). Suppose X is a k × n real matrix such that XA and XU have no zero entries. Then, A U B = sgn(XA) sgn(XU) is a full (n + k) × (n + k) PN pattern and, thus, it is a SAP. 64 Chapter 2. Sign Pattern Matrices

2.3.6 Some Open Problems Concerning SAPs Here are a few open problems related to SAP’s. In 2009, Catral–Olesky–van-den- Driessche [8] gave a survey of allow problems concerning spectra of sign patterns, and concluded with a list of open problems mostly related to SAPs. Several of these are still open and some are included below. (i) Find new techniques for proving that a sign pattern is a SAP. Current techniques require the knowledge of a nilpotent realization or explicit calculations with the characteristic polynomial; for a method that uses the Intermediate Value Theorem to avoid ﬁnding an explicit nilpotent realization, see Cavers– Kim–Shader–Vander-Meulen [12]. (ii) Are there other families of sign patterns (besides star sign patterns and 4 × 4 tree sign patterns) for which potential nilpotency and potential stability imply that the family is spectrally arbitrary?

(iii) Is it possible that S1 ⊕ S2 is a SAP with neither S1 nor S2 a SAP (cf., Example 2.3.13)? (iv) Prove (or disprove!) the 2n conjecture; see Conjecture 2.3.11. (v) Find techniques for constructing PN patterns. This would also be useful in the study of SAPs as both the Nilpotent-Jacobian method (Theorem 2.3.6) and the nilpotent-centralizer method (Theorem 2.3.8) depend on a nilpotent realization.

2.4 Reﬁned Inertia of Sign Patterns

2.4.1 Definition and Maximum Number of Refined Inertias Recall from Section 2.3.1 that the inertia of a matrix A is a triple of nonnegative integers, giving the number of eigenvalues of A with positive, negative, and zero real parts. Motivated by classes of sign patterns relevant for dynamical systems, Kim et al. [41] introduced the following definition, which refines the inertia by subdividing those eigenvalues with zero real part. Noting that for a real matrix complex eigenvalues occur in pairs, the refined inertia of a real n × n matrix A is the 4-tuple of nonnegative integers ri(A) = (n+, n−, nz, 2np), in which (counting multiplicities) n+ is the number of eigenvalues with positive real parts, n− is the number of eigenvalues with negative real parts, nz is the number of zero eigenvalues, and 2np is the number of nonzero pure imaginary eigenvalues. It follows that n+ + n− + nz + 2np = n, and the inertia of A is the triple given by i(A) = (n+, n−, nz + 2np). The refined inertia of a sign pattern S is {ri(A) | A ∈ Q(S)}. For an n × n sign pattern S, the refined inertia of S can be related to potential stability and sign stability. If S allows refined inertia (0, n, 0, 0), then S is potentially stable, 2.4. Refined Inertia of Sign Patterns 65

whereas if S requires refined inertia (0, n, 0, 0), then S is sign stable. If nz = 0, i.e., det(A) =6 0 for all A ∈ Q(S), then S is sign nonsingular (SNS). So, a SNS pattern has all realizations with determinant the same nonzero sign. For n ≥ 2, the maximum number of distinct inertias allowed by any n × n sign pattern is easily seen to be (n+1)(n+2)/2. The maximum number of distinct refined inertias allowed by any n × n sign pattern can be found by induction using this result and considering separately the cases with np = 0 and np > 0. Theorem 2.4.1 (Deaett–Olesky–van-den-Driessche, [18]). The maximum number R(n) of distinct refined inertias allowed by an n × n sign pattern with n ≥ 2 is R(n) = (k + 1)(k + 2)(4k + 3)/6 for n = 2k, and R(n) = (k + 1)(k + 2)(4k + 9)/6 for n = 2k + 1.

2.4.2 The Set of Refined Inertias Hn In Section 2.2.2, the community matrix of a dynamical system was considered. The magnitudes of matrix entries may be unknown but the signs known (for example a grass-rabbit-fox system: rabbits eat grass, foxes eat rabbits), so the matrix should be regarded as a sign pattern. The location of eigenvalues of the community matrix determine the stability of the underlying dynamical system. The community matrix of the grass-rabbit-fox system given in section 2.3.1 is sign stable; thus, it has refined inertia (0, 3, 0, 0). Consider now a different 3 × 3 community matrix. Example 2.4.2. Let

 − + 0   −1 1 0  S =  + − +  and A =  2 −2 1  . 0 − 0 0 −2 0

Sign pattern S is potentially stable as A ∈ Q(S) has refined inertia (0, 3, 0, 0). If the (2, 2) entry is changed to −1, then the resulting matrix has refined inertia (0, 1, 0, 2), with a pair of nonzero pure imaginary eigenvalues. If the (2, 2) entry is further changed to −0.5, then the resulting matrix has refined inertia (2, 1, 0, 0), and is unstable. Thus, by varying the magnitude of one entry (but retaining the sign), a pair of eigenvalues can cross the imaginary axis, and the dynamical system with this linearized sign pattern S may exhibit a Hopf bifurcation giving rise to periodic solutions. The three refined inertias allowed (and actually required) by the above sign pattern are motivation for the particular set of refined inertias that are next investigated. This set was introduced in Bodine–Deaett–McDonald–Olesky–van-den- Driessche [4]. For n ≥ 2, let Hn = {(0, n, 0, 0), (0, n − 2, 0, 2), (2, n − 2, 0, 0)}. An n×n sign pattern S requires (refined inertia) Hn if Hn = {ri(A) | A ∈ Q(S)}, and allows (refined inertia) Hn if Hn ⊆ {ri(A) | A ∈ Q(S)}. If S requires Hn, then S 66 Chapter 2. Sign Pattern Matrices

n allows Hn, and S is SNS with the sign of the determinant equal to (−1) for all A ∈ Q(S). Results are now given that address the problem of identifying n×n irreducible sign patterns that require or allow Hn, and then the results are used to give some examples from applications to detect whether or not Hopf bifurcation giving rise to periodic solutions may occur in some dynamical systems. Many of the results are taken from Bodine et al. [4], where examples are given of irreducible sign patterns that require Hn for 3 ≤ n ≤ 7. Recently, Gao–Li–Zhang [28] identiﬁed three families of n × n star sign patterns that require Hn for all n ≥ 4.

2.4.3 Sign Patterns of Order 3 and H3 Theorem 2.4.3. If S is an irreducible 3 × 3 sign nonsingular pattern that allows H3, then S requires H3. For the proof note that, by Theorem 2.4.1, the maximum number of distinct refined inertias for n = 3 is R(3) = 13. Since S is SNS and allows H3, det(A) < 0 for all A ∈ Q(S), thus nz = 0. Of the six refined inertias having nz = 0, three have positive determinant, and the remaining three are H3. Olesky–Rempel–van-den-Driessche [47] used an exhaustive search to list all refined inertias of tree sign patterns of order 3, three of which require H3. Bodine et al. [4] listed general 3 × 3 sign patterns that require H3, and the list was completed by Garnett–Olesky–van-den-Driessche [30], who also proved the following characterization. Theorem 2.4.4. Let S be an irreducible 3 × 3 sign pattern. Then, the following are equivalent:

(i) S requires H3; (ii) S is potentially stable, sign nonsingular but not sign stable; and (iii) S requires negative determinant and allows reﬁned inertia (0,1,0,2). In the proof it is shown that no irreducible 3 × 3 sign pattern S can have ri(S) = {0, 3, 0, 0), (0, 1, 0, 2)}. This result should be compared with the 5 × 5 sign pattern S in Example 2.2.3 that has ri(S) = {(0, 5, 0, 0, ), (0, 3, 0, 2)}.

2.4.4 Sign Patterns of Order 4 and H4 Theorem 2.4.5. If S is an irreducible 4 × 4 sign nonsingular pattern that requires a negative trace and allows H4, then S requires H4. The proof follows from noting that, from Theorem 2.4.1, a sign pattern of order 4 can allow at most 22 reﬁned inertias, but only those in H4 are consistent with both a negative trace and a positive determinant (as required by the sign pattern being sign nonsingular and potentially stable). As an example, consider a companion matrix sign pattern. 2.4. Reﬁned Inertia of Sign Patterns 67

Example 2.4.6. Let  0 + 0 0   0 1 0 0   0 0 + 0   0 0 1 0  B =   with B =   ∈ Q(B),  0 0 0 +   0 0 0 1  − − − − −d −c −b −a where a, b, c, d > 0. By varying the magnitudes of a, b, c, d, the sign pattern B allows all characteristic polynomials with all coeﬃcients positive. In particular, it allows the following polynomials: (z +1)4 giving (0, 4, 0, 0) ∈ ri(B), (z2 +1)(z +1)2 giving (0, 2, 0, 2) ∈ ri(B), and (z2 − z + 8)(z + 2)2 giving (2, 2, 0, 0) ∈ ri(B). Thus, B allows H4 and, since it is SNS and requires a negative trace, it requires H4. For tree sign patterns of order 4 (paths or stars), Garnett–Olesky–van-den- Driessche [29] started with the list of potentially stable tree sign patterns in Johnson–Summers [40] and in Lin–Olesky–van-den-Driessche [43] to prove the following characterization.

Theorem 2.4.7. A 4 × 4 tree sign pattern requires H4 if and only if it is potentially stable, sign nonsingular, and not sign stable, and its negative is not potentially stable. These conditions are not suﬃcient for a tree sign pattern of order 5 to require H5, as shown by Example 2.2.3.

2.4.5 Sign Patterns with All Diagonal Entries Negative The first two results below apply to sign patterns having all diagonal entries nonzero and follow from continuity since the two half planes are open. Lemma 2.4.8. If S is a sign pattern with all its diagonal entries nonzero that allows refined inertia (a, b, c, d), then it allows refined inertias (a + c + d, b, 0, 0) and (a, b + c + d, 0, 0). Corollary 2.4.9. An n × n sign pattern with all its diagonal entries nonzero allows Hn if and only if it allows refined inertia (0, n − 2, 0, 2). The previous corollary leads to the following useful result about patterns with all negative diagonal entries that allow Hn, and a way of constructing such patterns of higher orders. Let Im be the m × m sign pattern with each diagonal entry equal to + and all other entries 0.

Lemma 2.4.10. If Sn has all diagonal entries negative and allows Hn, then any superpattern of Sn allows Hn. Also, every superpattern of Sn ⊕−Im allows Hn+m.

For n ≥ 3, deﬁne Kn = Cn − In, where Cn = [cij] is the sign pattern of a negative n-cycle matrix with c12, c23, . . . , cn−1,n = + and cn1 = −, with all other entries 0. Thus, Kn has a signed digraph as illustrated in Fig. 2.4. The sign pattern Kn allows Hn, but requires Hn only for small values of n, as stated precisely in the next theorem. 68 Chapter 2. Sign Pattern Matrices

-−- − k k q+ - q 1 2J − J+

−kq JJ^ q k? − 6 JJ]n 3 +J + qn − 1 4 J q k... k − −

Figure 2.4: The signed digraph of Kn.

Theorem 2.4.11. The sign pattern Kn = Cn − In allows Hn for all n ≥ 3, but requires Hn if and only if 3 ≤ n ≤ 6.

This result is proved by ﬁrst observing that each An ∈ Q(Cn) has eigenvalues that are a positive scalar multiple of the nth roots of −1, so there is a unique pair of complex conjugate eigenvalues with maximum real part α > 0. The matrix An−αIn ∈ Q(Kn) has ri(An−αIn) = (0, n−2, 0, 2). Applying Corollary 2.4.9 gives that Kn allows Hn. For the require statement, diﬀerent values of n are considered. (i) For n = 3 and 4, the result follows from previous theorems as the sign pattern is SNS and requires a negative trace.

(ii) For n = 5, K5 requires negative trace and negative determinant so ri(K5) ⊆ (H5 ∪ {(4, 1, 0, 0), (2, 1, 0, 2), (0, 1, 0, 4)}). To show that K5 does not allow reﬁned inertia (4, 1, 0, 0) take a general matrix A5 ∈ Q(K5), and form its characteristic polynomial, the Routh Hurwitz matrix, and its leading principal minors. Counting sign changes in these minors gives n+(A) ≤ 2. Then, Lemma 2.4.8 shows that K5 also does not allow reﬁned inertias (2, 1, 0, 2) and (0, 1, 0, 4). (iii) For n = 6, the argument is similar to that for n = 5.

(iv) For n ≥ 7, take A ∈ Q(Kn) with magnitude 1 for every nonzero entry on the cycle Cn and magnitude < cos(3π/n) on the diagonal. Then, A = Cn − In has n+ ≥ 4, so Kn does not require Hn.

2.4.6 Detecting Periodic Solutions in Dynamical Systems Results on reﬁned inertia can be useful in detecting the possibility of Hopf bifurcation in dynamical systems when entries of the Jacobian (community) matrix are known only up to signs or may have some additional magnitude restrictions. The following general result is from Culos–Olesky–van-den-Driessche [17]. Recall that sgn(J) denotes the sign pattern of matrix J. 2.4. Reﬁned Inertia of Sign Patterns 69

Theorem 2.4.12. Let J be the n × n Jacobian matrix of a differential equation system evaluated at a steady state x∗, and depending on a vector of parameters p with each entry of J having fixed sign. (i) If sgn(J) requires refined inertia (0, n, 0, 0), then x∗ is linearly stable for all p.

(ii) If sgn(J) does not allow Hn, then the system does not have periodic solutions around x∗ arising from a Hopf bifurcation. (iii) If the entries of J have (have no) magnitude restrictions and if the restricted (unrestricted) sign pattern sgn(J) allows Hn, then the system may give rise to a Hopf bifurcation at a certain value of vector p.

Note that if sgn(J) allows Hn, then an additional condition is needed to determine whether or not these bifurcating periodic solutions about x∗ are linearly stable. The grass-rabbit-fox system is an example of the ﬁrst and second cases of this theorem. Some examples of the third case are now given. Example 2.4.13 (Goodwin Model). A model for a regulatory mechanism in cellular physiology is formulated as a system of three ordinary diﬀerential equations: dM V dE dP = − aM, = bM − cE, = dE − eP. dt K + P m dt dt Here, M, E, and P respectively represent the concentrations of messenger RNA, the enzyme, and the product of the reaction of the enzyme and a substrate, with the other letters being positive parameters. Linearizing about a steady state (with P at its steady state) gives the Jacobian matrix

 V mP m−1    −a 0 − (K+P m)2 − 0 − J =  b −c 0  with sgn(J) =  + − 0  , 0 d −e 0 + − which is equivalent to K3. Since equivalent sign patterns have the same set of refined inertias, this sign pattern requires H3. Periodic solutions are found numerically to occur for this Goodwin model for certain parameter values. To illustrate this, Fig. 2.5 shows plots of P against time with all parameters except b fixed. In the top plot b = 0.15 and P approaches the steady state. In the bottom plot, b = 0.5 and P oscillates about the steady state. Example 2.4.14 (Lorenz System). Consider an example with some magnitude restrictions. This comes from a 3-dimensional differential equation system introduced by Lorenz in 1963 to model the motion of a fluid layer. The equations linearized about one of the nonzero steady states give the Jacobian matrix  −σ σ 0   − + 0  J =  1 −1 −γ  with sgn(J) =  + − −  , γ γ −b + + − 70 Chapter 2. Sign Pattern Matrices

1.79

P 1.78

1.77 2.5 3 3.5 4 4.5 5 Time x 104 2.6 2.4 2.2 P 2 1.8

2.5 3 3.5 4 4.5 5 Time x 104

Figure 2.5: Plots of P against time with all parameters except b fixed. Top plot: b = 0.15; bottom plot: b = 0.5. and parameters σ > 0, b > 0, r > 1, and γ = (b(r − 1))1/2. This sign pattern is equivalent to a superpattern of K3, so the sign pattern allows H3 (see Lemma 2.4.10) but does not require H3 as it is not SNS. But the entries in J have some magnitude restrictions that need to be taken into account. For σ > b + 1 taking r as the bifurcation parameter and defining r0 = σ(σ + b + 3)/(σ − b − 1), the refined inertia of J has the following values: if r < r0, then ri(J) = (0, 3, 0, 0), if r = r0, then ri(J) = (0, 1, 0, 2), and if r > r0, then ri(J) = (2, 1, 0, 0). As r increases through r0, the Lorenz system undergoes a Hopf bifurcation and periodic orbits arise that may be stable or unstable depending on the values of σ and b and the nonlinear terms. Fig. 2.6 depicts solutions of the Lorenz attractor for parameter values σ = 10, b = 8/3, and r = 28 > r0. Example 2.4.15 (Infectious Disease Model). A constant population is divided into three disjoint classes with S(t), I(t), R(t) denoting the fractions of the population that are susceptible to, infectious with, and recovered from a disease. Positive parameters β and γ denote the constant contact rate and recovery rate, respectively. Assume that the disease confers temporary immunity on recovery (e.g., influenza). This is modeled by splitting R(t) into a chain of recovered classes R1,R2,...,Rk with the waiting time in each subclass assumed exponentially distributed with mean waiting time 1/. The S, I, R1, R2, ..., Rk, S model has flowchart 2.4. Refined Inertia of Sign Patterns 71

Figure 2.6: Lorenz attractor: σ = 10, b = 8/3, r = 28 > r0.

This system has a disease free steady state with S = 1 and other variables zero. If β < γ, then this is the only steady state and the disease dies out. If β > γ, then there is also an endemic (positive) steady state with S∗ = γ/β, ∗ ∗ ∗ I = (1 − γ/β)/(1 + kγ/), and Ri = γI /. To determine the stability of this endemic steady state, consider the Jacobian matrix at this steady state. Taking for example three recovered classes (k = 3) and writing S = 1 − I − R1 − R2 − R3, we have  −βI∗ −βI∗ −βI∗ −βI∗   − − − −   γ − 0 0   + − 0 0  J =   with sgn(J) =   .  0 − 0   0 + − 0  0 0 − 0 0 + −

This gives another example with magnitude restrictions in which the sign pattern (a superpattern of K4) allows H4. The leading principal submatrices of J of orders 2, 3, 4 give the Jacobian with k = 1, 2, 3 recovered classes, as now considered:

(i) k = 1, an S, I, R1, S model. The leading 2 × 2 subpattern of sgn(J) requires reﬁned inertia (0,2,0,0), and is sign stable.

(ii) k = 2, an S, I, R1, R2, S model. The leading 3 × 3 subpattern of sgn(J) allows H3, but the magnitude structure in J restricts its reﬁned inertia to (0,3,0,0), so this model is stable.

(iii) k = 3, an S, I, R1, R2, R3, S model. Here, sgn(J) allows refined inertia H4, and different parameter values in J give refined inertias (0, 4, 0, 0), (0, 2, 0, 2), and (2, 2, 0, 0). This model exhibits stable periodic solutions for some parameter values; see Hethcote–Stech–van-den-Driessche [35]. 72 Chapter 2. Sign Pattern Matrices

Example 2.4.16 (Three Competitor System). A three-species competition model, as considered by Takeuchi [50], has each species competing in a common patch X and also having its own refuge patch Yi, for species i = 1, 2, 3. Let xi be the population size competing in patch X and yi be the population size in refuge Yi. The time evolution is described by the following system of six ordinary diﬀerential equations.

dx1 = x1r1(1 − x1 − α2x2 − β3x3) + 1(y1 − x1), dt dx2 = x2r2(1 − β1x1 − x2 − α3x3) + 2(y2 − x2), dt dx3 = x3r3(1 − α1x1 − β2x2 − x3) + 3(y3 − x3), dt dy1 = y1R1(1 − y1) + 1(x1 − y1), dt dy2 = y2R2(1 − y2) + 2(x2 − y2), dt dy3 = y3R3(1 − y3) + 3(x3 − y3). dt

Positive parameters αi, βi are competition coeﬃcients, i is the dispersal rate for species i between patch X and patch Yi, and Ri and ri are intrinsic growth rates of species i in its refuge patch and competition patch. Assuming that there exists a ∗ ∗ positive steady state xi , yi , then the Jacobian matrix around this has sign pattern NI S = 3 , (2.1) I3 −I3 where N is the 3-by-3 sign pattern with every entry negative. Sign pattern N allows H3 (as it is equivalent to a superpattern of K3). Thus, S allows H6 since it is a superpattern of N ⊕ −I3. Takeuchi [50] shows numerically that the system has periodic solutions for speciﬁc parameter values.

2.4.7 Some Open Problems Concerning Hn

The set Hn is relatively new, so there are open problems concerning this set and its applications. Here are a few:

(i) Characterize sign patterns requiring Hn for n ≥ 4. This is currently known only for tree sign patterns with n = 4 (see Garnett et al. [29]). (ii) How can sign patterns that allow purely imaginary (nonzero) eigenvalues be identiﬁed? Digraph conditions would be very helpful.

(iii) How can sign patterns that do not allow Hn be identiﬁed? Such a result would be useful in applications. 2.5. Inertially Arbitrary Sign Patterns 73

(iv) Develop a theory for sign patterns with some magnitude restrictions, such as often occur in applications.

2.5 Inertially Arbitrary Sign Patterns

2.5.1 Definition and Relation to Other Properties From the definition given in section 2.3.1, an n × n sign pattern S is an inertially arbitrary sign pattern (IAP) if i(S) contains every triple of nonnegative integers (n+, n−, n0) with n+ + n− + n0 = n. In a similar way, an n × n sign pattern S is a refined inertially arbitrary pattern (rIAP) if ri(S) contains every ordered 4- tuple (n+, n−, nz, 2np) with n+ + n− + nz + 2np = n. Pattern S is a minimal IAP (rIAP) if no proper subpattern is an IAP (rIAP). One obvious method of showing that S is an IAP is, for each possible inertia, find an explicit matrix realization of S having this inertia, but this becomes impractical as n increases. Necessary conditions for an IAP stated in terms of the directed graph of the sign pattern are given in the following theorem due to Cavers–Vander-Meulen [13]. Theorem 2.5.1. If a sign pattern S is an inertially arbitrary pattern, then D(S) has at least one positive and one negative 1-cycle, and a negative 2-cycle. The result for the 1-cycle is obvious, since the trace must be allowed to have any sign. For the 2-cycle, assume that A = [aij] ∈ Q(S) has i(A) = (0, 0, n) giving tr(A) = 0. If ib` for ` = 1, . . . , m are the nonzero eigenvalues of A, then the n−2 Pm 2 characteristic polynomial of A has z coefficient E2 = b ≥ 0. But E2 is P k=1 k k 0. From 2 k

If S is a reﬁned inertially arbitrary sign pattern, then S is potentially nilpotent; however, if S is inertially arbitrary this implication may not be true. Example 2.5.3. By Kim et al. [42]

 − − − 0 0   + + + 0 0    G5 =  0 0 0 − −     0 − 0 0 −  − 0 0 0 0 is an IAP that is not potentially nilpotent. Setting every nonzero entry equal to 1 except the (2, 1) entry set to 2 gives a matrix with reﬁned inertia (0,0,3,2), thus inertia (0,0,5).

2.5.2 Generalization of the Nilpotent-Jacobian Method A generalization of the Nilpotent-Jacobian method as stated in Theorem 2.3.6 is useful in deriving results about inertia and refined inertia. To state this, the definition of a matrix allowing a nonzero Jacobian was introduced by Cavers et al. [11]. If the n × n matrix A = [aij] satisfies the following conditions then A allows a nonzero Jacobian: (i) A has m ≥ n nonzero entries; and

(ii) among these m nonzero entries there are n nonzero entries, say ai1j1 ,...,

ainjn , such that if X is the matrix obtained from A by replacing these entries n n−1 with x1, . . . , xn and pX (z) = z +p1z +···+pn−1z+pn is the characteristic polynomial of X, then the n × n Jacobian matrix with (i, j) entry equal to ∂pi (x1, . . . , xn) is nonsingular at (x1, . . . , xn) = (ai j , . . . , ai j ). ∂xj 1 1 n n The Nilpotent-Jacobian method can then be restated; see Cavers et al. [11, Th. 3.1]. Theorem 2.5.4. Let A be a nilpotent realization of a sign pattern S such that A allows a nonzero Jacobian. Then, every superpattern of S is spectrally arbitrary. By replacing the nilpotent matrix in the Nilpotent-Jacobian method by a realization that allows a nonzero Jacobian, Cavers et al., proved the following result. Theorem 2.5.5 (Cavers–Garnett–Kim–Olesky–van-den-Driessche–Vander-Meulen, [11, Thm. 3.2]). Let S be an n × n sign pattern with A ∈ Q(S) allowing a nonzero n n−1 Jacobian. If the characteristic polynomial of A is cA(z) = z + a1z + ··· + an−1z+an then, for any real b1, . . . , bn suﬃciently close to a1, . . . , an, respectively, each superpattern Sˆ of S has a realization B ∈ Q(Sˆ) with characteristic polynomial n n−1 cB(z) = z + b1z + ··· + bn−1z + bn. 2.5. Inertially Arbitrary Sign Patterns 75

Some results from Cavers et al.,[11] are now given to illustrate the usefulness of this generalization. Lemma 2.5.6. Let S be an n × n sign pattern with superpattern Sˆ. Suppose A ∈ Q(S) allows a nonzero Jacobian and ri(A) = (n+, n−, nz, 2np).

(i) if nz ≥ 1, then (n+ +1, n−, nz −1, 2np) and (n+, n− +1, nz −1, 2np) ∈ ri(Sˆ);

(ii) if nz ≥ 2, then (n+, n−, nz − 2, 2(np + 1)) ∈ ri(Sˆ); and

(iii) if np ≥ 1, then (n+ + 2, n−, nz, 2(np − 1)) and (n+, n− + 2, nz, 2(np − 1)) ∈ ri(Sˆ). Moreover, for each modified refined inertia there is a realization of Sˆ with this refined inertia that allows a nonzero Jacobian. Note that the condition that A allows a nonzero Jacobian is necessary for the statement of this modification lemma. This can be seen from the 5×5 sign pattern given in Example 2.2.3 that has a realization A with ri(A) = (0, 3, 0, 2) but does not allow a nonzero Jacobian. This sign pattern is sign semi-stable, so it does not allow refined inertia (2,3,0,0). The above lemma can be applied recursively to an initial refined inertia to yield a set of allowed inertias. Corollary 2.5.7. Let S be an n × n sign pattern with superpattern Sˆ. Suppose A ∈ Q(S) has ri(A) = (n+, n−, nz, 2np) with nz ≥ 2. If A allows a nonzero Jacobian then, for every m+ ≥ n+, m− ≥ n− and m0 ≥ 0 with m+ +m− +m0 = n, it follows that (m+, m−, m0) ∈ i(Sˆ). For example, if A ∈ Q(S) has ri(A) = (1, 4, 3, 2) and allows a nonzero Jaco- bian, then (2, 7, 1), (4, 6, 0) ∈ i(Sˆ). A sufficient condition for a sign pattern to be inertially arbitrary, first proved by Cavers–Fallat [10, Thm. 2.13], follows from the above corollary, and illustrates the use of refined inertia. Theorem 2.5.8. Let S be an n × n sign pattern with A ∈ Q(S) having ri(A) = (0, 0, nz, 2np) for some nz ≥ 0. If A allows a nonzero Jacobian, then every superpattern of S is inertially arbitrary. To end this subsection, a centralizer technique is stated that gives a sufficient condition for a sign pattern to be an IAP; see Cavers et al.,[11, Theorem 4.9]. Theorem 2.5.9. Let S be an n × n sign pattern with A ∈ Q(S) having ri(A) = (0, 0, nz, 2np) with nz ≥ 2. If every eigenvalue of A has geometric multiplicity one and the only matrix B in the centralizer of A satisfying B ◦ AT = 0 is the zero matrix, then every superpattern of S is inertially arbitrary.

If nz = n in the above theorem, then A is nilpotent and every superpattern is a SAP; see Theorem 2.3.8. However, if nz < n, then this may not be true; for example, G5 does not allow reﬁned inertia (0, 0, 5, 0) but allows reﬁned inertia (0, 0, 3, 2). 76 Chapter 2. Sign Pattern Matrices

2.5.3 Reducible IAPs

If S1 and S2 are IAPs, then clearly S1 ⊕ S2 is an IAP. However, this result is not in general true if IAP is replaced by rIAP, since the direct sum of two odd-order sign patterns requires at least two real eigenvalues. The question whether S1 ⊕ S2 can be an IAP with neither summand an IAP has been addressed by Cavers [9] with the following example. Example 2.5.10 (Cavers, [9]). + + + 0  − − − 0  B4 =   + 0 0 + + 0 0 0 is not an IAP but B4 ⊕ T2 is an IAP, where T2 is the SAP (and hence an IAP) given in Example 2.3.4. Moreover, B4 ⊕ (−B4) is a smallest example of an IAP with both summands not IAPs, but B4 ⊕ (−B4) is not PN, so it is not a SAP.

2.5.4 A Glimpse at Zero-Nonzero Patterns A zero-nonzero pattern is a matrix with entries from {0, ∗}. A sign pattern gives a unique corresponding zero-nonzero pattern. Many of the previous definitions and ideas given for sign patterns carry over to zero-nonzero patterns. Results have been given for these patterns, for example, n = 4 IAPs by Cavers–Vander- Meulen [14], and n = 4 SAPs by Corpuz–McDonald [16]. Some inclusions hold; for example, the refined inertia of a sign pattern is a subset of the refined inertia of the corresponding zero-nonzero pattern. Note that matrices A and −A have the same zero-nonzero pattern, thus if S is a zero-nonzero pattern that allows refined inertia (n+, n−, nz, 2np), then it allows its reversal refined inertia (n−, n+, nz, 2np). Thus, in counting the number of possible refined inertias for a zero-nonzero pattern, it is best to exclude reversals. Taking the total number of refined inertias of a sign pattern as given by R(n) in Theorem 2.4.1 and subtracting the number of refined inertias in which the first two coordinates are both equal, the next result was proved by Deaett–Olesky–van-den-Driessche [18]. Theorem 2.5.11. The maximum number R∗ of distinct refined inertias excluding reversals allowed by any n × n zero-nonzero pattern with n ≥ 2 is R∗(n) = (n + 2)(n + 3)(n + 4)/24 for n even, and R∗(n) = (n + 1)(n + 3)(n + 5)/24 for n odd. Relationships between some spectral properties are now summarized: (i) for n = 2 and 3, IAP, rIAP, and SAP zero-nonzero patterns are identical; (ii) for n ≤ 4, an n × n zero-nonzero pattern is an rIAP if and only if it is a SAP; (iii) for n = 4, there are examples of zero-nonzero patterns that are IAPs but not rIAPs; for example, the zero-nonzero pattern N ∗ corresponding to the sign pattern N given in Example 2.5.2 is an IAP but is not an rIAP. 2.5. Inertially Arbitrary Sign Patterns 77

(iv) For n = 5, there exists an irreducible zero-nonzero pattern L given below that is an rIAP but not a SAP. Note that the existence of such an example is currently unknown for sign patterns. Example 2.5.12 (Deaett–Olesky–van-den-Driessche, [18]).

 ∗ ∗ 0 0 ∗   0 0 ∗ 0 ∗    L =  0 0 0 ∗ 0     0 0 ∗ 0 ∗  ∗ ∗ 0 0 ∗

5 4 3 2 is an rIAP but not a SAP. If p(z) = z + p1z + p2z + p3z + p4z + p5, then it is shown that L fails to allow characteristic polynomial p(z) if and only if p1 = p3 = 0 while p5 =6 0. Thus, L is not a SAP. However, L allows any p(z) of the form (z − 1)k(z + 1)5−k for k ∈ {0, 1,..., 5} and also any p(z) with a zero on the imaginary axis; thus, L is an rIAP. The pattern L and the zero-nonzero pattern M corresponding to the sign pattern given by Example 2.3.13 give a 9 × 9 reducible pattern L ⊕ M that is a SAP with neither summand a SAP. It is currently unknown whether such an example exists for sign patterns.

2.5.5 A Taste of More General Patterns Cavers–Fallat [10] consider allow problems with a larger set of symbols S = {0, +, −, +0, −0, ∗, #}, where +0, −0 represent a nonnegative resp. nonpositive real number, ∗ represents a nonzero real number, and # represents an arbitrary real number. An S-pattern is a matrix with symbols in S. Definitions of inertially arbitrary etc. carry over to S-patterns but the different symbols, especially #, allow more flexibility. Example 2.5.13. Consider a 4 × 4 companion matrix with symbols 0, +, and # given by  0 + 0 0   0 0 + 0  C4 =   .  0 0 0 +  ####

Then, C4 is a SAP, IAP, rIAP and is PN. Note that D(C4) allows a positive loop, allows a negative loop, and allows a negative 2-cycle, but does not allow two oppositely signed loops. These same properties hold for an n × n companion matrix S-pattern giving SAPs with 2n − 1 nonzero entries (cf., the 2n conjecture for sign pattern matrices). Recall that, for a sign or zero-nonzero pattern, the implications in general are that if S is a SAP, then it is an rIAP, which implies that it is an IAP, in 78 Chapter 2. Sign Pattern Matrices turn implying that it is potentially stable; and, in addition, if S is an rIAP, then it is potentially nilpotent. However, if S is an n × n {+, −, ∗, #}-pattern, then it is a SAP if and only if it is an rIAP, if and only if it is potentially nilpotent. The proofs of these statements for S are modiﬁcations of Pereira’s proof [48]; see Theorem 2.3.12.

2.5.6 Some Open Problems Concerning IAPs Some open problems for IAPs are now stated, some of which are applicable to both sign patterns and zero-nonzero patterns: (i) If a sign pattern S is an rIAP, then is it necessarily a SAP? (ii) Determine the minimum number of nonzero entries in an n × n IAP for sign patterns and for zero-nonzero patterns; see Cavers–Vander-Meulen–Vander- spek [15], where it is shown that such a pattern can have fewer than 2n nonzero entries.

(iii) Determine conditions on i(S1) and i(S2) ensuring that certain inertias are in i(S1 ⊕ S2) for sign patterns and for zero-nonzero patterns. (iv) Determine any other families (besides star sign patterns) for which the sets of IAPs and SAPs are identical. Bibliography

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Spectral Radius of Graphs by Dragan Stevanovi´c

Graphs are naturally associated with matrices, as matrices provide a simple way of representing graphs in computer memory. The basic one of these is the adjacency matrix, which encodes existence of edges joining vertices of a graph. Knowledge of spectral properties of the adjacency matrix is often useful to describe graph properties which are related to the density of the graph’s edges, on either a global or a local level. For example, entries of the principal eigenvector of adjacency matrices have been used in the study of complex networks, introduced under the name eigenvector centrality by the renowned mathematical sociologist Phillip Bonacich back in 1972; see [5, 6]. More recently, Van-Mieghem–Omi´c– Kooij [53] have shown that the epidemic threshold in SIS-type network infections is equal to the reciprocal value of the spectral radius of the network’s adjacency matrix. Our focus in this chapter is on mathematical properties of the spectral radius of the adjacency matrix of a simple graph. Most often these properties will be upper or lower bounds that relate the spectral radius to other properties of a graph. Our main goal here is to illustrate diﬀerent approaches used to prove such results in the literature.

3.1 Graph-Theoretical Deﬁnitions

Let G = (V,E) be a graph with the vertex set V having n = |V | vertices, and the edge set E having m = |E| edges. In this chapter we assume G to be a simple graph, meaning that each edge joins two distinct vertices, there is at most one edge 84 Chapter 3. Spectral Radius of Graphs

Figure 3.1: An example of a simple graph. joining any given pair of distinct vertices, and edges have no orientation. Thus, we may identify an edge e ∈ E with the two-element subset {u, v} of V consisting of the vertices it joins. Instead of repeatedly writing e = {u, v}, we will denote this just as e = uv. When uv ∈ E we also say that the vertices u and v are adjacent, or that u and v are neighbors. The set of all neighbors of a vertex u in a graph G is denoted by NG(u). Here and afterwards we will drop the subscript G whenever the graph is clear from the context. The number of neighbors of the vertex u is its degree and is denoted by deg(u). The maximum and the minimum vertex degree of G are denoted by ∆ and δ, respectively. A graph G is called d-regular (or just regular) if all vertices have degree d. A walk in the graph G is a sequence W : w0, . . . , wk of its vertices, such that wi and wi+1 are adjacent in G for each i = 0, . . . , k − 1. The vertex w0 is the starting vertex of W , wk is the ending vertex, and k is the length of W . Note that a walk, in general, may contain both repeated vertices and repeated edges. A graph G is connected if there exists a walk between any pair of its vertices. There is no need to include the word “distinct” in the previous sentence, as W : u is an example of a trivial walk of length zero starting and ending at any given vertex u. The distance d(u, v) between two vertices u and v is the length of the shortest walk between them, if any walk between them exists; otherwise it is set to ∞. The eccentricity ecc(u) of a vertex u is the distance to the vertex farthest from u in G. The diameter D and the radius r of a connected graph G are, respectively, the maximum and the minimum eccentricity of its vertices. Hence the diameter is the distance between the two farthest vertices, while the radius corresponds to the central vertices of G. A graph H = (V 0,E0) is a subgraph of G = (V,E) if both V 0 ⊆ V and E0 ⊆ E. H is a spanning subgraph of G if V 0 = V . On the other hand, H is an induced subgraph of G if E0 contains all the edges of E joining the vertices of V 0. Example 3.1.1. Let us brieﬂy illustrate the above deﬁnitions on the graph shown in Fig. 3.1. This graph has the vertex set V = {a, b, c, d, e, f}, and the edge set is E = {ab, ad, ae, af, bc, bd, cd}. The neighbors of vertex a form its neighborhood N(a) = {b, d, e, f}, so that the degree of a is four. This is also the maximum 3.2. The Adjacency Matrix and Its Spectral Properties 85

Figure 3.2: Examples of a complete, complete bipartite, and complete multipartite graph, a cycle, a path, and a star. vertex degree in the graph, so that ∆ = 4. On the other hand, δ = 1 as both e and f have smallest degrees in this graph. W : e, a, b, d, c, b, d, a, f is a walk of length eight between e and f. However, the distance between e and f is just two, as the shortest walk between them is W 0 : e, a, f. The eccentricities of vertices c, e, and f are equal to three, as (c, e) and (c, f) are pairs of farthest vertices at distance three. The eccentricities of vertices a, b, and d are equal to two, as any other vertex may be reached from them using at most two edges. Thus the diameter is D = 3 and the radius is r = 2. The graph ({a, b, c, d}, {ab, bc, cd}) is a subgraph of this graph, but it is neither spanning nor induced. An example of a spanning subgraph is ({a, b, c, d, e, f}, {ab, bc, cd, ae, af}), while an example of an induced subgraph is ({a, b, c, d}, {ab, ad, bc, bd, cd}). Let us now deﬁne several types of graphs that will be used throughout this chapter. The complete graph Kn has vertices 1, . . . , n and all edges ij for 1 ≤ i < j ≤ n. The complete multipartite graph Kn1,...,np , p ≥ 2, is formed from the union of disjoint vertex sets V1,...,Vp, such that |Vi| = ni and it contains an edge uv whenever u and v belong to diﬀerent vertex sets. For p = 2 the graph Kn1,n2 is also called the complete bipartite graph. The cycle Cn, n ≥ 3, has vertices 1, . . . , n and edges {i, i + 1 | i = 1, . . . , n − 1} ∪ {1, n}. A graph is said to be a tree if it is connected and it does not contain a cycle as a subgraph. Types of trees that appear frequently throughout the chapter include paths and stars. The path Pn has vertices 1, . . . , n and edges {i, i + 1 | i = 1, . . . , n − 1} (hence it is a cycle with one edge deleted). The star Sn has n vertices, one of which is a central vertex that is adjacent to the remaining n − 1 vertices. Examples of graphs of these types are illustrated in Fig. 3.2.

3.2 The Adjacency Matrix and Its Spectral Properties

The adjacency matrix A of a simple graph G = (V,E) is a square matrix indexed by the set of vertices V such that ( 1, if uv ∈ E, Au,v = (3.1) 0, if uv∈ / E. 86 Chapter 3. Spectral Radius of Graphs

While the adjacency matrix encodes the existence of edges between vertices of G, its powers encode the numbers of walks of a given length, as visible from the following classical theorem. Theorem 3.2.1. For k ≥ 0 and vertices u and v of G, the number of walks of length k k starting at u and ending at v is equal to (A )u,v. Proof. By induction on k. For k = 0, there exists a single walk W : u of length zero between u and v if and only if u = v, which corresponds to the entry of 0 (A )u,v = Iu,v, which is equal to one if and only if u = v. Assuming that the theorem has already been proved for some k ≥ 0, the proof for k + 1 follows by observing that each walk of length k + 1 between u and v consists on a walk of length k between u and some neighbor w of v, followed by the edge wv. The number of all walks of length k + 1 between u and v is thus equal to X k X k k+1 (A )u,w = (A )u,wAw,v = (A )u,v, w∈N(v) w∈V where the change of summation range in the second sum is supported by the deﬁnition (3.1) of the adjacency matrix. The fact that the adjacency matrix A of a simple graph is real and symmetric has two consequences. First, if G has n = |V | vertices, then A has real eigenvalues λ1 ≥ · · · ≥ λn that are the roots of the characteristic polynomial P (λ) = det(λI − A). (3.2)

Secondly, A can be diagonalized, i.e., there exist eigenvectors x1, . . . , xn of A that n form an orthonormal basis of R , where xi, i = 1, . . . , n, satisﬁes the eigenvalue equation λixi = Axi. Due to (3.1), this eigenvalue equation may also be written as X (∀u ∈ V ) λixi,u = xi,v. (3.3) v∈N(u)

The orthonormal eigenvectors x1, . . . , xn provide the spectral decomposition n X > A = λixixi . (3.4) i=1 Pn > To see why (3.4) holds, let B = i=1 λixixi . Then, for each j = 1, . . . , n, n X > Axj − Bxj = λjxj − λixixi xj = λjxj − λjxj = 0, (3.5) i=1 > > due to orthonormality of eigenvectors: xi xj = 0 for i =6 j and xj xj = 1. Let V (ev)v∈V be the standard basis of R , deﬁned by ( 1, if u = v, ev,u = 0, if u =6 v. 3.2. The Adjacency Matrix and Its Spectral Properties 87

V Since the eigenvectors x1, . . . , xn form a basis of R , there exist real coeﬃcients Pn cv,j, v ∈ V , j = 1, . . . , n, such that ev = j=1 cv,jxj. Then, from (3.5), we have Pn > > Aev −Bev = j=1 cv,j (Axj − Bxj) = 0, so that ﬁnally Au,v = eu Aev = eu Bev = Bu,v. The adjacency matrix of a connected graph is also irreducible, which facili- tates the application of the Perron–Frobenius theorem (see, e.g., Gantmacher [25, Ch. XIII] for a proof). Theorem 3.2.2. An irreducible, nonnegative n × n matrix A has a real, positive eigenvalue λ1 such that:

(i) λ1 is a simple eigenvalue of A;

(ii) λ1 has a positive eigenvector x1; and

(iii) |λi| ≤ λ1 holds for all other (possibly complex) eigenvalues λi, i = 2, . . . , n.

In addition, if A has a total of h eigenvalues whose moduli are equal to λ1, then the spectrum of A is symmetric with respect to the rotation for the angle 2π/h in the complex plane.

Thus the largest eigenvalue λ1 of the adjacency matrix A is also its spectral radius. The corresponding positive unit eigenvector x1 is called the principal eigenvector. Many of the results that follow in this chapter will strongly depend on positivity of the principal eigenvector. The simplest among them is the bound

λ1 ≤ ∆, (3.6) which follows from the eigenvalue equation (3.3) by focusing on the vertex v with the maximum principal eigenvector component x1,v = maxu∈V x1,u: X λ1x1,v = x1,u ≤ deg(v) x1,v ≤ ∆ x1,v. u∈N(v)

However, the fact that the spectral radius is a simple eigenvalue is also useful. If we > happen to know a positive eigenvector y of A, then y x1 > 0 so that y cannot be orthogonal to the principal eigenvector x1, hence y has to be itself an eigenvector that corresponds to the spectral radius of A. Theorem 3.2.3. The spectral radius of a d-regular graph is equal to d. Proof. If a graph G = (V,E) is d-regular, then the all-ones vector 1 is a positive eigenvector of its adjacency matrix corresponding to the eigenvalue d: X X (∀u ∈ V ) d1u = d = 1 = 1v. v∈N(u) v∈N(u)

By the Perron–Frobenius theorem the spectral radius of G equals d. 88 Chapter 3. Spectral Radius of Graphs

We will now brieﬂy describe current approaches used to prove results on the spectral radius of the adjacency matrix. They will be worked out in more detail in subsequent sections. The ﬁrst approach uses the principal eigenvector components to deduce properties of λ1. It is usually based on the Rayleigh quotient characterization of the spectral radius: x>Ax λ1 = sup > . (3.7) x6=0 x x Pn Namely, if x is represented as x = i=1 cixi in the orthonormal eigenvector basis, then

> Pn Pn > Pn Pn > Pn 2 x Ax cicj xi Axj cicj λj xi xj λ c i=1 j=1 i=1 j=1 i=1 i i ≤ > = Pn Pn > = Pn Pn > = Pn 2 λ1, x x i=1 j=1 cicj xi xj i=1 j=1 cicj xi xj i=1 ci with equality if and only if x = c1x1 with c1 =6 0. Since the Rayleigh quotient represents a supremum over all nonzero vectors, it allows deducing a lower bound x>Ax for λ1 by estimating x>x for a suitable vector x. For example, by taking x = 1 we obtain > P 1 A1 u,v∈V Au,v 2m λ1 ≥ = = , (3.8) 1>1 n n where n is the number of vertices and m the number of edges, showing that λ1 is bounded from below by the average vertex degree of G. Further examples of the principal eigenvector approach are showcased in Section 3.4. The second approach enables comparison of spectral radii of two graphs G and H by comparing the values of their characteristic polynomials. Suppose that we have somehow managed (e.g., by using reduction procedures given later, in Section 3.5) to show that

(∀λ ≥ l) PG(λ) > PH (λ). (3.9)

Now, if l happens to be the spectral radius λH,1 of H, the above inequality implies that the spectral radius λG,1 of G is strictly smaller than λH,1. Namely, the characteristic polynomial of the adjacency matrix is a monic polynomial, so that limλ→+∞ PH (λ) = +∞ and, for λ greater than or equal to the largest root λH,1 of PH (λ), we have PH (λ) ≥ 0. Together with (3.9) this implies that PG(λ) > 0 for all λ ≥ λH,1, showing that the largest root λG,1 of PG(λ) is then necessarily smaller than λH,1. A few detailed examples of this approach will be given in Section 3.5. The third approach also enables the comparison of spectral radii of two graphs, this time by comparing the numbers of (closed) walks of any given length in these graphs. Let Nk denote the number of all walks of length k in G, and let Mk denote the number of closed walks of length k. From the spectral decomposition k (3.4) and the fact that x1, . . . , xn are also the eigenvectors of A with eigenval- k k k Pn k > ues λ1 , . . . , λn, we have A = i=1 λi xixi , which, together with Theorem 3.2.1, 3.3. The Big Gun Approach 89 implies that n !2 X k X k X Nk = (A )u,v = λi xi,u , (3.10) u,v∈V i=1 u∈V

n ! n X k X k X 2 X k Mk = (A )u,u = λi xi,u = λi . (3.11) u∈V i=1 u∈V i=1

Being expressible as the sums of powers of eigenvalues, the values Mk are also called spectral moments in the literature. From these equations it is not hard to see that, for a connected graph G,

pk λ1 = lim Nk, (3.12) k→∞ 2pk λ1 = lim M2k. (3.13) k→∞ The proof of these equalities will be given in Section 3.6. The appearance of 2k in the second equality is justified by the fact that if G is bipartite, meaning that its vertex set may be divided into two parts such that the end vertices of each edge belong to different parts, then G does not contain closed walks of odd length, so that M2k+1 = 0 for each k ≥ 0. Now, if we somehow manage to show for two graphs G and H that for infinitely many values of k either NG,k ≤ NH,k or MG,k ≤ MH,k then (3.12) or (3.13) implies that λG,1 ≤ λH,1. Inequality between the numbers of (closed) walks is usually shown by providing an injective mapping from the set of (closed) walks of G into the set of (closed) walks of H. This approach will be discussed in much more detail in Section 3.6. Usually only one of these approaches will be suitable for a particular problem, but we will see that a few of the forthcoming results are elegantly proved in at least two ways. However, before we delve deeper into them, we will take a short note of one more, aptly named, approach.

3.3 The Big Gun Approach

In situations when we are not able to deduce any sufficiently strong claim about the graph’s principal eigenvector, characteristic polynomial, or numbers of walks, it may pay off to rely on a few well-known theorems that, instead of focusing just on the spectral radius, offer wider information about the whole spectrum of a graph. Actually, trying to apply the following theorems is usually the first (but not necessarily successful) step in tackling any problem about the spectral radius. Theorem 3.3.1 (Interlacing Theorem). Let A be a real symmetric n × n matrix with eigenvalues λ1 ≥ · · · ≥ λn, and let B be an arbitrary principal submatrix of A obtained by deleting k rows and k columns of A with the same indices. If the eigenvalues of B are µ1 ≥ · · · ≥ µn−k, then

λi ≥ µi ≥ λi+k, i = 1, . . . , n − k. (3.14) 90 Chapter 3. Spectral Radius of Graphs

Proof. We begin by extending the Rayleigh quotient characterization of (3.7) to other eigenvalues of A. Let x1, . . . , xn denote the orthonormal eigenvector basis of A, such that xi is an eigenvector corresponding to λi, i = 1, . . . , n. Let l ≤ n and Pl suppose that x ∈ hx1, . . . , xli. Then, x = i=1 cixi for some real cis and

> Pl Pl > Pl Pl > Pl 2 x Ax i=1 j=1 cicj xi Axj i=1 j=1 cicj λj xi xj i=1 λici = = = ≥ λl, x>x Pl Pl > Pl Pl > Pl 2 i=1 j=1 cicj xi xj i=1 j=1 cicj xi xj i=1 ci (3.15) ⊥ with equality if and only if x = clxl. On the other hand, if x ∈ hx1, . . . , xl−1i , Pn then x = i=l dixi for some real di’s and

> Pn Pn > Pn Pn > Pn 2 x Ax didj xi Axj didj λj xi xj λ d i=l j=l i=l j=l i=l i i ≤ > = Pn Pn > = Pn Pn > = Pn 2 λl, x x i=l j=l didj xi xj i=l j=l didj xi xj i=l di (3.16) with equality if and only if x = dlxl. Now, let y1, . . . , yn−k be the orthonormal eigenvector basis of B, such that yi is an eigenvector corresponding to µi, i = 1, . . . , n − k. Further, let x1|B, . . . , xn|B denote the vectors obtained by restricting x1, . . . , xn to the indices of B. Let n−k i ≤ n − k. Since hy1, . . . , yii spans a subspace of R of dimension i, while ⊥ hx1|B, . . . , xi−1|Bi spans a subspace of dimension at least (n − k) − (i − 1), there ⊥ exists a nonzero vector s ∈ hy1, . . . , yii ∩ hx1|B, . . . , xi−1|Bi . Let sA be the vector obtained from s by appending zeros as the components for the indices that were ⊥ deleted from A. Then sAxi = sxi|B, so that sA ∈ hx1, . . . , xi−1i . Thus,

> > sA A sA s Bs λi ≥ > = > ≥ µi. sA sA s s

Finally, µi ≥ λi+k is obtained by applying the above reasoning to −A and −B. The previous theorem is a classical result whose proof goes back to Cauchy, which is why it is sometimes called the Cauchy interlacing theorem. Among graphs, the adjacency matrix of H is a principal submatrix of the adjacency matrix of G when H is an induced subgraph of G. If G is connected and H is a proper subgraph of G, then the strict inequality λ1 > µ1 holds in (3.14). Two classical examples of the use of the Interlacing Theorem among graphs follow.

Theorem 3.3.2 (Smith, [50]). Connected graphs with λ1 ≤ 2 are precisely the induced subgraphs of the graphs shown in Fig. 3.3.

Proof. Vertices of each graph in Fig. 3.3 are labeled with components of positive eigenvectors corresponding to the eigenvalue 2, showing that λ1 = 2 for each of these graphs. Now, let G be a connected graph with λG,1 ≤ 2. From 2m/n ≤ λ1 we get that m ≤ n, showing that G is either a tree or a unicyclic graph. 3.3. The Big Gun Approach 91

Figure 3.3: The Smith graphs with λ1 = 2.

If G is unicyclic, then it contains a cycle Ck for some k as an induced subgraph. If Ck is a proper subgraph of G, then λG,1 > λCk,1 = 2, and we have a contradiction. Hence, G must be the cycle Cn itself. If G is√ a tree, then it contains a star K1,∆ as its induced subgraph, and since λK1,∆,1 = ∆, we see that ∆ ≤ 4. If ∆ = 4, then from λK1,4,1 = 2 we see that G must be the star K1,4 itself. If ∆ = 3, then G cannot contain more than two vertices of degree 3, as otherwise some Wk would be a proper induced subgraph of G and we would have λG,1 > λWk,1 = 2. Moreover, if G contains exactly two vertices of degree 3, then G must be equal to Wn itself. Suppose, therefore, that G contains exactly one vertex of degree 3. If each of three paths attached to this vertex has length at least 2, then G must be F7. Otherwise, if one of the attached paths has length 1 and the other two have length at least 3, then G must be F8. In the remaining case, G must be an induced subgraph of either F9 or Wk for some k. Finally, if ∆ = 2, then G is a path and as such is an induced subgraph of some cycle from Fig. 3.3. Theorem 3.3.3 (Smith, [50]). If a connected graph G with at least two vertices has exactly one positive eigenvalue, then G is either a complete graph or a complete multipartite graph. Proof. The statement is easily checked for connected graphs with either two or three vertices, as the only such graphs are P2, P3, and K3. Assume, therefore, that G = (V,E) contains at least four vertices and that it is neither a complete nor a complete multipartite graph. Then G contains three vertices u, v, and w, such that uv ∈ E, but uw, vw∈ / E. Since G is connected, it contains a vertex z adjacent to w. Then, depending on the existence of edges uz and vz in G, its subgraph induced by the vertices {u, v, w, z} is isomorphic to one of the four graphs depicted in Fig. 3.4. The second largest eigenvalue in each 92 Chapter 3. Spectral Radius of Graphs

Figure 3.4: Possible induced subgraphs from Theorem 3.3.3. of these four graphs is positive, so that by the Interlacing Theorem, the second largest eigenvalue of G has to be positive as well.

The previous theorem holds in the other direction as well. However, that part of the proof requires describing the full set of eigenvectors of complete multipartite graphs in order to estimate all of their eigenvalues, which would turn our attention away from the spectral radius. For details of the proof in the other direction see, for example, Stevanovi´c[51, Sect. 3.4]. Another form of interlacing is given in the following theorem. Let A be a square matrix indexed by the set V and let π : V = V1 ∪ · · · ∪ Vk be an arbitrary partition of the index set. For i, j = 1, . . . , k, let AVi,Vj be the submatrix of A with rows in Vi and columns in Vj, and let bi,j denote the average row sum of

AVi,Vj . The matrix B = (bi,j) is called the quotient matrix of A corresponding to the partition π. Theorem 3.3.4 (Haemers, [29]). Let A be a real symmetric matrix, and let B be its quotient matrix. If A has eigenvalues λ1 ≥ · · · ≥ λn and B has eigenvalues µ1 ≥ · · · ≥ µk, then λi ≥ µi ≥ λi+n−k, for i = 1, . . . , k. As a matter of fact, Haemers [29] has obtained the previous theorem as a corollary of a general interlacing theorem, which also includes Theorem 3.3.1 as a special case. For details see either Brouwer–Haemers [8, Sect. 2.5] or Haemers [29]. The use of interlacing of the eigenvalues of an adjacency matrix with the eigenvalues of its quotient matrices is illustrated in the following theorem. A set of vertices S is independent in G if there are no edges in G joining vertices of S. The independence number α is the largest size of an independent set in G.

Theorem 3.3.5 (Haemers, [29]). Let G be a simple graph with λ1 and λn denoting the largest and the smallest eigenvalue of the adjacency matrix of G, respectively. Then, its independence number α satisﬁes

λ1λn α ≤ n 2 , (3.17) λ1λn − δ where δ is the minimum vertex degree of G. 3.3. The Big Gun Approach 93

Proof. Let S be an independent set of G of size α. Partition the vertex set V as V = S ∪ (V − S). If m denotes the number of edges in G, and k the average degree of vertices in S, then the quotient matrix of the adjacency matrix of G corresponding to this partition is equal to

0 k B = kα 2(m−kα) . n−α n−α

If µ1 ≥ µ2 are the eigenvalues of B, then Theorem 3.3.4 yields

k2α δ2α −λ1λn ≥ −µ1µ2 = − det(B) = ≥ , n − α n − α which implies (3.17). The following result localizes the spectrum of a complex matrix to the union of closed circles in the complex plane. Theorem 3.3.6 (Gerschgorin circle theorem, [26]). Let A be a complex n×n matrix, P and let rk = i6=k |Aki| be the sum of moduli of nondiagonal entries in the row k. Sn Then each eigenvalue λ is contained in the set k=1 C(Akk, rk), where C(Akk, rk) is the circle {x ∈ C | |x − Akk| ≤ rk}. Proof. Let λ be an eigenvalue of A with the corresponding eigenvector x. Let k be such that |xk| = maxi |xi|. From the eigenvalue equation λx = Ax, we have P (λ − Akk) xk = i6=k Akixi, so that X X |λ − Akk| |xk| = Akixi ≤ |Aki||xi| ≤ rk|xk|. i6=k i6=k

Division by |xk| > 0 now implies that λ ∈ C(Akk, rk). The Weyl inequalities represent another classical tool in the repertoire of a spectral graph theorist.

Theorem 3.3.7 (Weyl). Let A and B be n × n Hermitian matrices. Let λ1 ≥ · · · ≥ λn be the eigenvalues of A, µ1 ≥ · · · ≥ µn the eigenvalues of B, and ν1 ≥ · · · ≥ νn the eigenvalues of A + B. If k + l ≤ n + 1, then νk+l−1 ≤ λk + µl, while if k + l ≥ n + 1, then λk + µl ≤ νk+l−n. Proof. Similarly to the Interlacing Theorem, the proof of Weyl inequalities relies on the Rayleigh quotient characterizations (3.15) and (3.16) and nonempty intersection of corresponding vector subspaces. Let x1, . . . , xn be the orthonormal eigenvectors of A, y1, . . . , yn the orthonormal eigenvectors of B, and z1, . . . , zn the orthonormal eigenvectors of A + B. ⊥ The subspace X = hx1, . . . , xk−1i has dimension n − k + 1, the subspace Y = ⊥ hy1, . . . , yl−1i has dimension n − l + 1, while the subspace Z = hz1, . . . , zk+l−1i has dimension k + l − 1. The dimension of the intersection X ∩ Z is at least 94 Chapter 3. Spectral Radius of Graphs

(n − k + 1) + (k + l − 1) − n = l. Since the dimension of Y is n − l + 1, we conclude that there exists a nonzero vector v ∈ X ∩ Y ∩ Z so that, from (3.15) and (3.16), v>(A + B)v v>Av v>Bv νk+l−1 ≤ = + ≤ λk + µl. v>v v>v v>v A similar argument is applied in the case k + l ≥ n + 1. The Weyl inequalities enable one to decompose a graph as the union of edge- disjoint spanning subgraphs and then work with each of these subgraphs separately. After obtaining upper bounds for the spectral radii of spanning subgraphs, the Weyl inequalities represent the final step that provides an upper bound for the whole graph. For example, Dvo˘rák–Mohar[21] have used this method to obtain an upper bound for the spectral radius of planar graphs. A graph is planar if it can be drawn in a plane such that its edges do not cross each other. Theorem 3.3.8 (Dvo˘rák–Mohar,[21]). If G is a planar graph with the maximum p √ vertex degree ∆ ≥ 2, then λ1 ≤ 2 2(∆ − 2) + 2 3.

For ∆ ≤ 3, the theorem follows from λ1 ≤ ∆, while for 4 ≤ ∆ the proof relies on the decomposition result of Gon¸calves: Theorem 3.3.9 (Gon¸calves, [28]). If G is a planar graph, then there exist forests

T1, T2, and T3 such that E(T ) = E(T1) ∪ E(T2) ∪ E(T3) and ∆T3 ≤ 4.

The edges of T1 can be oriented in such a way that at most one edge points away from any given vertex: choose a root in each component of T1 and orient all edges toward the root. The same can be done for T2, so that their union T1 ∪ T2 gives a graph in which edges are oriented so that at most two edges point away from any given vertex. Hayes [31] has proved the following bound on the spectral radius of such graphs. Theorem 3.3.10 (Hayes, [31]). Let G be a simple graph with the maximum vertex degree ∆, whose edges can be oriented such that at most d edges point away from p any given vertex with d ≤ ∆/2. Then, λ1 ≤ 2 d(∆ − d). p From Hayes’s theorem we therefore have λT1∪T2,1 ≤ 2 2(∆ − 2). Edges of the forest T3 can also be oriented so that at most one edge points away from any given vertex, and another application of the Hayes theorem yields λT ,1 ≤ p √ 3 2 1(∆ − 1) ≤ 2 3. The Weyl inequalities now provide the cumulative bound p √ λG,1 ≤ λT1∪T2,1 + λT3,1 ≤ 2 2(∆ − 2) + 2 3. However, it is necessary to note here that general results like the Interlacing Theorem or the Weyl inequalities can rarely provide a sharp estimate of the spectral radius of a graph. For example, while Theorem 3.3.8 might be useful for planar graphs with small vertex degrees, it does not yield much when attempted against the following conjecture on the maximum spectral radius of a planar graph. Inter- estingly, this conjecture appeared independently in a geographical journal in 1991 and in a mathematical journal in 1993. 3.4. The Eigenvector Approach 95

Conjecture 3.3.11 (Boots–Royle, [7]; Cao–Vince [10]). If G is a planar graph with n vertices then λG,1 ≤ λK2∨Pn−2,1, where K2 ∨ Pn−2 is a graph obtained from the path Pn−2 and the complete graph K2 by joining the two vertices of K2 to each vertex of Pn−2.

Since the maximum vertex degree of K2 ∨ Pn−2 is ∆ = n − 1, Theorem 3.3.8 p √ applied to K2 ∨ Pn−2 yields λK2∨Pn−2,1 ≤ 2 2(n − 3) + 2 3, while, representing the edge set of K2 ∨ Pn−2 as the union of stars K1,n−1 and K1,n−2 and the path Pn−2, the Interlacing Theorem and the Weyl inequalities yield that √ √ √ n − 1 < λK2∨Pn−2,1 < n − 1 + n − 2 + 2.

When applied to arbitrary planar graphs, Theorem 3.3.8 does imply Conjec- ture 3.3.11, but only for those planar graphs satisfying n + 27 1p ∆ ≤ − 3(n − 1). 8 2 A related conjecture slightly predating Conjecture 3.3.11 concerns the maximum spectral radius of outerplanar graphs. An outerplanar graph is a planar graph in which one face (the outer face) contains all vertices of the graph. Cvetkovićand Rowlinson posed the following conjecture in 1990: Conjecture 3.3.12 (Cvetković–Rowlinson, [19]). If G is an outerplanar graph with n vertices, then λG,1 ≤ λK1∨Pn−1,1, with K1 ∨ Pn−1 denoting a graph obtained from the path Pn−1 by adding a new vertex adjacent to each vertex of Pn−1. Both of these conjectures are still open. The biggest obstruction to their solution seems to be the absence of results on the change in spectral radius after local modifications that preserve planarity of a graph.

3.4 The Eigenvector Approach

The Rayleigh quotient (3.7) has, as its consequences, a number of lemmas describing the change in spectral radius after simple local modiﬁcations. The recipe for showing λG,1 ≥ λH,1 using the Rayleigh quotient is simple: start with the positive principal eigenvector xH,1 of the graph H and, with AG and AH denoting the adjacency matrices of G and H respectively, show that

> > xH,1AGxH,1 xH,1AGxH,1 > ≥ > = λH,1. xH,1xH,1 xH,1xH,1

The spectral radius λG,1 is then, as a supremum of such quotients, also greater than or equal to λH,1. The case of equality can be easily dealt with, since it holds if and only if xH,1 is a principal eigenvector of G as well.

Lemma 3.4.1. If G is a simple graph and e is an edge not in G, then λG+e,1 > λG,1. 96 Chapter 3. Spectral Radius of Graphs

Proof. Let e = uv and let x1 be a positive principal eigenvector of G. Then,

> > > x1 AG+ex1 x1 AGx1 2x1,ux1,v x1 AGx1 > = > + > > > = λG,1, x1 x1 x1 x1 x1 x1 x1 x1

> x AG+ex due to x1,ux1,v > 0. The lemma now follows from λG+e,1 = supx6=0 x>x . This lemma shows that the spectral radius is monotone with respect to addition of edges. Consequently, among simple graphs on n vertices the complete graph Kn has the maximum spectral radius, while the graph with no edges Kn has the minimum spectral radius. Lemma 3.4.2 (Rowlinson, [46]). Let p, q, and r be the vertices of a simple graph G = (V,E) such that pq ∈ E and pr∈ / E. If the principal eigenvector x1 of G satisﬁes x1,q ≤ x1,r, then λG−pq+pr,1 > λG,1. Proof. We have

> > x1 AG−pq+prx1 x1 AGx1 −2x1,px1,q + 2x1,px1,r λG−pq+pr,1 ≥ > = > + > ≥ λG,1, x1 x1 x1 x1 x1 x1 due to x1,q ≤ x1,r. The equality λG−pq+pr,1 = λG,1 cannot hold as then x1 would also be the principal eigenvector of G − pq + pr and the eigenvalue equation at the vertex q in G and G − pq + pr, X λG,1x1,q = x1,u

u∈NG(q) X X λG,1x1,q = x1,u = x1,u − x1,p,

u∈NG−pq+pr (q) u∈NG(q) would imply x1,p = 0, which is contradictory to the positivity of x1. Remark 3.4.3. It is important to notice that, after the application of this edge rotation lemma, the inequality x1,q ≤ x1,r holds for the principal eigenvector of the new graph G−pq+pr as well. Otherwise, if x1,q > x1,r would hold then the spectral radius would further increase by deleting the edge pr from G − pq + pr and adding the edge pq, resulting in contradictory inequalities λG,1 < λG−pq+pr,1 < λG,1. This shows that we can apply the lemma simultaneously to a sequence of vertices p1, . . . , pk to obtain λG−p1q−···−pkq+p1r+···+pkr,1 > λG,1 if x1,q ≤ x1,r. The edge rotation lemma enables one to compare spectral radii of graphs with the same number of edges. For example, it may be used to prove one part of a classical result.

Theorem 3.4.4. If T is a tree on n vertices, then λT,1 ≤ λSn,1, with equality if and ∼ only if T = Sn. 3.4. The Eigenvector Approach 97

Proof. If T is not a star, then T contains two vertices u and v, both adjacent to some leaves. Let u1, . . . , uk be the leaves of T adjacent to u, and let v1, . . . , vl be the leaves adjacent to v. Let x1 be the principal eigenvector of T . If x1,u ≤ x1,v then, by the edge rotation lemma, λT,1 < λT −u1u−···−uku+u1v+···+ukv, while if x1,u > x1,v then λT,1 < λT −v1v−···−vlv+v1u+···+vlu. In either case T does not have maximal spectral radius among trees, and moreover, both trees T −u1u−· · ·−uku+ u1v +···+ukv and T −v1v −· · ·−vlv +v1u+···+vlu have fewer vertices adjacent to leaves, indicating that repeated application of the above procedure necessarily results in the star Sn as the tree with the maximum spectral radius.

Unfortunately, edge rotation cannot be used to prove the other part of this ∼ classical result: that λPn,1 ≤ λT,1 with equality if and only if T = Pn. The point is that if x1,q > x1,r then one cannot conclude that λG−pq+pr,1 < λG,1, but only

> x1 AGx1 −2x1,px1,q + 2x1,px1,r λG−pq+pr,1 ≥ > + > < λG,1, x1 x1 x1 x1 which does not give any information on the relationship between λG−pq+pr,1 and λG,1. We need to resort to either the characteristic polynomial or the walk numbers in order to prove λPn,1 ≤ λT,1, and two diﬀerent proofs of this fact will be given in the next two sections. Rowlinson [47] had used the Rayleigh quotient to prove a special case of Con- jecture 3.3.12 for maximal outerplanar graphs without branching triangles. First, if an outerplanar graph has an internal face that is not a triangle, then its spectral radius can be increased by adding a chord of that face. Consequently, maximal spectral radius among outerplanar graphs is attained by a graph representing some triangulation of an outer n-gon. A branching triangle of a maximal outerplanar graph G is a triangular face none of whose edges belong to the outer n-gon.

Theorem 3.4.5 (Rowlinson, [47]). If G is a maximal outerplanar graph with n vertices, n ≥ 4, and no branching triangles, then λG,1 ≤ λK1∨Pn−1,1.

Proof. We may assume that n ≥ 6, as the only maximal outerplanar graphs without branching triangles on four and ﬁve vertices are exactly K1 ∨ P3 and K1 ∨ P4. Denote the vertices of G as 1, . . . , n in the order they appear on the outer n-gon of G. If G is not K1 ∨ Pn−1, then vertices can be labeled so that G contains triangles {1, 2, k} and {1, k, k+1} (see Fig. 3.5, left) and, without loss of generality, 0 we may suppose that x1,1 ≥ x1,k. Let G be the graph informally obtained by twisting a part of G on one side of the edge 1k; see Fig. 3.5, right. More formally, let i1, . . . , il be the neighbors of the vertex k in G having labels between 3 and k−1. 0 0 The graph G is then obtained as G = G−ki1 −· · ·−kil +1i1 +···+1il. Multiple application of the edge rotation Lemma 3.4.2 implies that λG0,1 > λG,1, so that a maximal outerplanar graph without branching triangles, other than K1 ∨ Pn−1, cannot have the maximum spectral radius. 98 Chapter 3. Spectral Radius of Graphs

Figure 3.5: The triangles from the proof of Theorem 3.4.5.

The Cauchy–Schwartz inequality is certainly one of the most useful inequalities in mathematics. A ﬁne example of its use with the principal eigenvector of a graph is found in the proof of Yuan Hong’s well-known bound.

Theorem 3.4.6 (Hong,√ [33]). Let G be a connected simple graph with n vertices and m edges. Then, λ1 ≤ 2m − n + 1, with equality if and only if G is isomorphic to the star Sn or the complete graph Kn.

Proof. Let x1 be the principal eigenvector of the adjacency matrix A of G = (V,E). −u For each vertex u ∈ V let x1 be the vector deﬁned as ( x , if uv ∈ E, x−u = 1,v 1,v 0, if uv∈ / E.

Let Au denote the u-th row of A. The eigenvalue equation at the vertex u gives P −u λ1x1,u = v∈N(u) x1,v = Aux1 . Having in mind that the row Au contains deg(u) entries equal to 1, we get   2 2 −u 2 2 −u 2 X 2 λ1x1,u = |Aux1 | ≤ |Au| |x1 | = deg(u) 1 − x1,v . v∈ /N(u)

Summing this inequality for all u ∈ V , we obtain

2 X X 2 λ1 ≤ 2m − deg(u) x1,v u∈V v∈ /N(u) X 2 X X 2 = 2m − deg(u)x1,u − deg(u) x1,v u∈V u∈V v∈{ / u}∪N(u) X 2 X 2 X = 2m − deg(u)x1,u − x1,v deg(u) u∈V v∈V u/∈{v}∪N(v) 3.4. The Eigenvector Approach 99

X 2 X 2 X ≤ 2m − deg(u)x1,u − x1,v 1 u∈V v∈V u/∈{v}∪N(v) X 2 X 2 = 2m − deg(u)x1,u − x1,v(n − deg(v) − 1) u∈V v∈V X 2 = 2m − x1,u(n − 1) u∈V = 2m − n + 1.

Equality holds in the last inequality only if, for each v ∈ V , either {v} ∪ N(v) = V or each nonneighbor of v, diﬀerent from v, has degree 1. If there exists a vertex w of degree 1, adjacent to some vertex z, then all other vertices of G have degree 1 and have to be adjacent to z, due to connectedness of G. In such case G is a star Sn. Otherwise, if G has no vertices of degree 1, then each vertex of G has to be adjacent to all other vertices√ of G, so that G is a complete graph Kn. It is straightforward to check that λ1 = 2m − n + 1 holds for both Sn and Kn.

Another application of the Cauchy–Schwartz inequality appears in the proof of the next result. We already know that λ1 ≤ ∆, where ∆ is the maximum vertex degree, and that equality holds if and only if the graph is regular. The author of this chapter raised the question of how close to ∆ the spectral radius of a nonregular graph can be? The following bound builds upon the author’s initial answer.

Theorem 3.4.7 (Cioab˘a–Gregory–Nikiforov, [13]). If a graph G is not regular, then ∆ − λ1 > 1/(n(D + 1)), where D is the diameter of G.

Proof. Let x1 be the principal eigenvector of the adjacency matrix A of G = (V,E). > P Then, x1 Ax1 = 2 x1,ux1,v. Taking into account that x1 is a unit vector, uv∈E P from the Rayleigh quotient (3.7) we get λ1 = 2 uv∈E x1,ux1,v. Therefore,

X 2 X ∆ − λ1 = ∆ x1,u − 2 x1,ux1,v u∈V uv∈E X 2 X 2 X = (∆ − deg(u))x1,u + deg(u)x1,u − 2 x1,ux1,v u∈V u∈V uv∈E X 2 X 2 2 X = (∆ − deg(u))x1,u + x1,u + x1,v − 2 x1,ux1,v u∈V uv∈E uv∈E X 2 X 2 = (∆ − deg(u))x1,u + (x1,u − x1,v) . u∈V uv∈E

Now, let s (resp., t) be the vertex of G with the maximum (resp., minimum) principal eigenvector component x1,max (resp., x1,min). If P : s = p0, p1, . . . , pk = t is 100 Chapter 3. Spectral Radius of Graphs the shortest walk between s and t, then we have the following chain of inequalities:

k−1 X 2 X 2 (x1,u − x1,v) ≥ x1,pi − x1,pi+1 uv∈E i=0 k−1 ! 1 X ≥ x1,p − x1,p k i i+1 i=0

1 2 ≥ (x1,max − x1,min) . D These inequalities follow from the facts that P takes up a subset of edges of

E, the Cauchy–Schwartz inequality applied to the vector of diﬀerences (x1,pi − x1,pi+1 )0≤i≤k−1 and the all-one vector 1, and the fact that k ≤ D. Having further in mind that the graph is not regular, we have ∆ − deg(u) ≥ 1 for at least one 2 2 vertex u ∈ V , so that ∆−λ1 ≥ x1,min+(x1,max−x1,min) /D. The right-hand side of the above inequality is a quadratic function in x1,min. As such, its minimum value 2 is equal to x1,max/(D +1), attained for x1,min = x1,max/(D +1). The theorem now 2 follows from x1,max > 1/n, where the strict inequality holds as G is not regular √1 (and hence its principal eigenvector is not n 1).

Cioab˘a[11] later improved the above bound slightly to ∆ − λ1 > 1/nD, but at the expense of a much more complicated proof, which we will not cover here. We know from the Rayleigh quotient (3.7) that, after deletion of an edge uv from a connected graph G, it holds that λG,1 − 2x1,ux1,v ≤ λG−uv,1. Van- Mieghem et al. [54] gave a mix of approximative and empirical evidence that λG−uv,1 has minimum value if one deletes an edge uv with the maximum product x1,ux1,v. Note, however, that this is just an expectation supported by experimental evidence, and no hard results exist on this topic. Peter Rowlinson [46] has managed to prove the following upper bound on λG−uv,1.

Theorem 3.4.8 (Rowlinson, [46]). For a connected graph G, it holds λG−uv,1 < λG,1 + 1 + ε − γ, where

ε(ε + 1)(ε + 2) γ = 2 = λG,1 − λG,2. (x1,u − x1,v) + ε(2 + ε − 2x1,ux1,v)

2 If 2x1,ux1,v > (x1,u − x1,v) , then this bound improves the simple bound λG−uv,1 < λG,1; see also further discussion of algebraic theory of matrix perturbations in Cvetkovi´c–Rowlinson–Simi´c[20, Sect. 6.4]. Although most often used, the Rayleigh quotient is not the only tool available in the eigenvector approach. Ostrowski [44] provided the following simple lemma back in 1960. Lemma 3.4.9 (Ostrowski, [44]). Let A be a nonnegative, irreducible n × n matrix. Pn Let ri = j=1 Aij be the entry sum of the row i of A, and let ∆ = maxi ri and 3.4. The Eigenvector Approach 101

Figure 3.6: Principal eigenvector components along a pendant path of G ◦ Pk+1.

δ = mini ri be the maximum and minimum row sums of A, respectively. If λ1 and x1 are the spectral radius and the principal eigenvector of A, then r x ∆ λ ∆ 1,max ≥ max , 1 ≥ . x1,min λ1 δ δ

Proof. Let p and q be the indices such that rp = ∆ and rq = δ. Then from Pn the eigenvalue equations λ1x1,max ≥ λ1x1,p ≥ j=1 Apjx1,min = ∆x1,min and Pn λ1x1,min ≤ λ1x1,q ≤ j=1 Aqjx1,max = δx1,max, from which the ﬁrst inequality of the lemma follows. The second inequality follows from

x 2 ∆ λ ∆ 1,max ≥ · 1 = . x1,min λ1 δ δ

Cioab˘a–Gregory[12] and Zhang [56] provided some improvements upon Os- trowski’s inequality. Moreover, motivated by the results of the computer search on graphs with up to nine vertices, Cioab˘a–Gregory[12] have further conjectured that “among the connected graphs on n vertices the maximum ratio of x1,max/x1,min is always attained by some kite graph formed by identifying a vertex of a complete graph and an end vertex of a path”. This is a rather plausible conjecture since the principal eigenvector components decrease along a pendant path approximately inversely proportional to λ1 > 2, while the maximum possible density of edges in the complete subgraph ensures large value of λ1. However, in order to prove the conjecture one has to be able to simultaneously control changes in both x1,max and x1,min. Taking care of just the principal eigenvector components along a pendant path is a much easier task. Let G◦Pk+1 denote the graph obtained by identifying a vertex of a connected graph G with an end vertex of the path Pk+1. Let λ1 denote the spectral radius of G ◦ Pk+1 and assume that its principal eigenvector components along the path are denoted as x0, . . . , xk as in Fig. 3.6. The eigenvalue equation for the vertices of a path gives rise to the recurrent equation λ1xl = xl−1 + xl+1, l ≥ 1. (3.18) 102 Chapter 3. Spectral Radius of Graphs

Suppose that the value of x0 is kept ﬁxed, so that it can be taken as one boundary condition. The other boundary condition, determining the sequence x0, . . . , xk completely, is λ1xk = xk−1, which can be alternatively stated as xk+1 = 0, after adding to the path Pk+1 an imaginary vertex with the component xk+1 (drawn in gray in Fig 3.6). Roots of the characteristic equation corresponding to the recurrence (3.18) are p 2 λ1 λ1 − 4 t1,2 = . (3.19) 2

They are distinct if λ1 > 2 and satisfy t1t2 = 1. Denoting the larger root as t > 1 and the smaller root as t−1 < 1 and solving for the boundary conditions yields the principal eigenvector components x1, . . . , xk as

t2k+2−i − ti xi = x0 , i = 1, . . . , k. t2k+2 − 1 This expression for the principal eigenvector components along a pendant path becomes much more useful when we allow the pendant path to be infinite. The spectral theory of infinite, locally finite graphs was put forward in a series of papers by Bojan Mohar in the 1980s and the best starting point for independent study is the survey Mohar–Woess [42]. What we need here from this theory are the following two facts.

Theorem 3.4.10 (Mohar, [41]). Assume that the sequence of ﬁnite graphs (Gn)n≥1 converges to the graph G, meaning that each edge of G is contained in all but ﬁnitely many graphs from the sequence. Then, the spectral radius of G is equal to rG = limn→∞ λGn,1.

Further, if we are able to ﬁnd a positive eigenvector x of G with ﬁnite l2-norm, then the corresponding eigenvalue λ has to be equal to the spectral radius of G; see

Stevanović[51, Sect. 2.2]. These two facts enable us to determine limk→∞ λG◦Pk+1,1 as the spectral radius of λG◦P∞,1, provided that G has a simple form. If x is a positive eigenvector of G ◦ P∞ with eigenvalue λ, the boundary conditions for the recurrence relation (3.18) become the fixed value of x0 and the requirement that x has a finite l2-norm. This requirement enforces that in i −i the general solution for the components xi = At + Bt we must have A = 0, so that the eigenvector components along an infinite pendant path are simply −i xi = x0t , i ≥ 1. Note that even if several infinite paths are attached to the same vertex, eigenvector components along each of these paths satisfy the same expression above. The eigenvalue equation for the components of x at the vertices of G now determine the corresponding eigenvalue λ. As the first example, suppose that G consists of a single vertex to which d ≥ 3 infinite pendant paths have been attached. If x0 is the eigenvector component of this central vertex, then the eigenvector component of each of its neighbors −1 in the pendant paths is equal to x1 = x0t . The eigenvalue equation for the −1 −1 central vertex yields the condition λx0 = dx1 = dx0t , from which λ = dt . 3.4. The Eigenvector Approach 103 √ −1 −1 Substituting λ = t + t from (3.19) yields t = (d − √1)t , or t = d − 1, from which the spectral radius is equal to λ = t + t−1 = d/ d − 1. For the second example, suppose that G is a complete graph Kn, n ≥ 3, and that a single infinite pendant path has been attached at its vertex u, giving us an infinite kite graph; see Stevanović[51, Ex. 2.3]. Let x0 denote the eigenvector component of u. Due to the fact that we want the corresponding eigenvalue λ to be simple and due to the symmetry of the remaining vertices of Kn, we may denote by x−1 the common eigenvector component of the remaining vertices of Kn. The eigenvalue equation at u and at the remaining vertices of Kn gives a system

λx0 = (n − 1)x−1 + x1,

λx−1 = (n − 2)x−1 + x0.

−1 −1 Substituting x1 = x0t in the above system yields the condition λ − t (λ−n+ −1 2) = n−1, which, after further substituting λ = t+t , yields a quadratic√ equation t2 − (n − 2)t − (n − 2) = 0, whose root greater than one is t = (n − 2 + n2 − 4)/2. −1 From λ = t + t we get back that the spectral radius of the inﬁnite kite Kn ◦ P∞ is n − 3 n − 1 p λ = + n2 − 4. 2 2(n − 2) To conclude this section, let us mention yet another way to use the principal n eigenvector of a graph. For two vectors x, y ∈ R let us write x ≤ y if xi ≤ yi for each i = 1, . . . , n, and x < y if x ≤ y but x =6 y. Let A be a real symmetric n × n matrix with the largest eigenvalue λA,1. The Rayleigh quotient (3.7) implies that

y > 0 and Ay > µy =⇒ λA,1 > µ, (3.20) as y>Ay y>µy λA,1 ≥ > = µ. y>y y>y On the other hand, if A is irreducible then

y > 0 and Ay < µy =⇒ λA,1 < µ. (3.21)

> Namely, if x1 > 0 is the positive principal eigenvector of A, then x1 y > 0 and > > > λA,1 < µ follows from λA,1x1 y = x1 Ay < µx1 y. An easy application of these observations is contained in the following result of Simi´c[49].

Theorem 3.4.11 (Simi´c,[49]). Let N(w) = N1 ∪ N2, N1 ∩ N2 = ∅, N1,N2 =6 ∅, be a partition of the neighborhood of vertex w of the connected graph G. Let G0 be the graph obtained by splitting a vertex w: G0 is obtained by deleting w from G, and adding two new vertices w1 and w2 such that w1 is adjacent to vertices in N1, while w2 is adjacent to vertices in N2. Then, λG0,1 < λG,1. 104 Chapter 3. Spectral Radius of Graphs

Figure 3.7: The tree Tn which has spectral radius 2. Components of a vector proportional to its principal eigenvector are shown next to each vertex.

Proof. We may suppose that G0 is connected, as otherwise it consists of two connected components, each of which is a proper subgraph of G with spectral radius strictly less than λG,1. Let x1 be the principal eigenvector of G = (V,E), and let y be a new vector indexed by V − {w} ∪ {w1, w2} and deﬁned by ( x1,w, if v = w1 or v = w2, yv = x1,v, if v =6 w1, w2.

0 Since G is connected, vectors x1 and y both are positive. Let A be the adjacency 0 matrix of G . Then, if v =6 w1, w2

0 X (A y)v = x1,u = λG,1x1,v = λG,1yv, u∈N(v) since v cannot be adjacent to both w1 and w2. On the other hand, if v = wi, i = 1, 2, then

0 X X (A y)v = x1,u < x1,u = λG,1x1,w = λG,1yv,

u∈Ni u∈N(w)

0 since Ni is a proper subset of N(w). Thus, y > 0 and A y < λG,1y, implying that λG0,1 < λG,1 by (3.21). A less straightforward application of (3.20) and (3.21) is used to prove the following result from Hoffman–Smith [32] on the spectral radius of an edge-sub- divided graph. Let uv be an edge of the connected graph G and let Guv be the graph obtained by subdividing the edge uv, i.e., by deleting uv and adding two new edges uw, wv, where w is a new vertex. The spectral radius of Guv remains equal to λG,1 = 2 if G is isomorphic to the cycle Cn or to the tree Tn shown in Fig. 3.7, and uv is not a pendant edge of Tn. The internal path of G is defined in Hoffman–Smith [32] as a walk W : v0,..., vk+1 such that either 3.5. The Characteristic Polynomial Approach 105

(i) all vertices v0, . . . , vk+1 are distinct, k ≥ 0, deg(v0) ≥ 3, deg(v1) = ··· = deg(vk) = 2, and deg(vk+1) ≥ 3; or

(ii) the vertices v0, . . . , vk are distinct, k ≥ 1, vk+1 = v0, deg(v0) ≥ 3, and deg(v1) = ··· = deg(vk) = 2. Note that if the degrees of vertices u and v are both at least 3, then the edge uv is itself an internal path of G. As a consequence, if uv does not lie on an internal path of G, then either G is isomorphic to the cycle Cn or G is a proper subgraph of Guv, in which case λG,1 < λGuv ,1. Otherwise, we have Theorem 3.4.12 (Hoffman–Smith, [32]). If uv lies on an internal path of the connected graph G and G is not isomorphic to Tn, then λGuv ,1 < λG,1. The complete proof of this theorem can also be found in the easier-to-find monograph Cvetković–Rowlinson–Simić[20] as Theorem 3.2.3. Neumaier [43] obtained bounds on the spectral radius by constructing partial eigenvectors for a given eigenvalue λ. The vector x, indexed by the vertex set V of G, is called a partial λ-eigenvector for the vertex z ∈ V if xz =6 0 and the P eigenvalue equation λxu = (Ax)u = v∈N(u) xv holds for all vertices u ∈ V −{z}. The value P u∈N(z) xu ξz,λ = λ − (3.22) xz is called the λ-exitvalue, with respect to z. If ξz,λ = 0 then x is an eigenvector and λ is an eigenvalue of G. Neumaier [43] showed that, if λ is not an eigenvalue of G − z, then there exists a unique partial λ-eigenvector for z. What interests us, however, is the case when G is connected and the partial λ-eigenvector is positive. Note that (3.22) is equivalent to (Ax)z = (λ − ξz,λ)xz. Hence, if ξz,λ > 0 then (Ax)z < λxz and, together with (Ax)u = λxu for u =6 z, we have that Ax < λx. From (3.21) it follows that λA,1 < λ in such a case. Otherwise, if ξz,λ < 0, then (Ax)z > λxz, so Ax > λx. Then (3.20) implies that λA,1 > λ. The exit values were successfully used by Brouwer–Neumaier [9] to complete p √ the characterization of graphs with λ1 ≤ 2 + √5 ≈ 2.0582, and later by Woo– 3 Neumaier [55] to characterize graphs with λ1 ≤ 2 2 ≈ 2.1312.

3.5 The Characteristic Polynomial Approach

In Section 3.2 we showed that the inequality

(∀λ ≥ λH,1) PG(λ) > PH (λ) (3.23) implies that λG,1 < λH,1, because PH (λ) ≥ 0 when λ ≥ λH,1. For graphs G and H diﬀering only locally, the deduction of (3.23) usually starts with one of the following Schwenk’s lemmas. 106 Chapter 3. Spectral Radius of Graphs

Lemma 3.5.1 (Schwenk, [48]). Let G and H be two vertex-disjoint graphs and let u be a vertex of G and v a vertex of H. Denote by G(u − v)H the graph obtained by adding an edge between u and v. Then,

PG(u−v)H (λ) = PG(λ)PH (λ) − PG−u(λ)PH−v(λ). In the special case when H consists only of vertex v, the characteristic polynomial of G(u − v)v reduces to λPG(λ) − PG−u(λ). Lemma 3.5.2 (Schwenk, [48]). Let G and H be two vertex-disjoint graphs and let u be a vertex of G and v a vertex of H. Denote by G(u = v)H the graph obtained by identifying vertices u and v. Then,

PG(u=v)H (λ) = PG(λ)PH−v(λ) + PG−u(λ)PH (λ) − λPG−u(λ)PH−v(λ).

Lemma 3.5.3 (Schwenk, [48]). For a vertex v of the graph G, let Cv denote the set of all cycles of G containing v. Similarly, for an edge uv of G, let Cuv denote the set of all cycles of G containing uv. Then, X X PG(λ) = λPG−v(λ) − PG−v−w(λ) − 2 PG−C (λ)

w∈N(v) C∈Cv and X PG(λ) = PG−uv(λ) − PG−u−v(λ) − 2 PG−C (λ).

C∈Cuv One of the earliest and probably the most-used example of difference of characteristic polynomials appears in the seminal paper Li–Feng [38]. Lemma 3.5.4 (Li–Feng, [38]). Let u be a vertex of a nontrivial connected graph G and, for nonnegative integers k and l, let G(u; k, l) denote the graph obtained from G by attaching pendant paths of lengths k and l at u. If k ≥ l ≥ 1, then λG(u;k,l),1 > λG(u;k+1,l−1),1. Since the original paper [38] is written in Chinese, we adapt here the proof from Cvetković–Rowlinson–Simić[20, Sect. 6.2]. Proof. By removing the end vertices of different attached paths, Lemma 3.5.1 implies that PG(u;k,l)(λ) = λPG(u;k,l−1)(λ) − PG(u;k,l−2)(λ), for l ≥ 2, and also PG(u;k+1,l−1)(λ) = λPG(u;k,l−1)(λ) − PG(u;k−1,l−1)(λ), so that

PG(u;k,l)(λ) − PG(u;k+1,l−1)(λ) = PG(u;k−1,l−1)(λ) − PG(u;k,l−2)(λ), l ≥ 2. Iterating this equality further yields

PG(u;k,l)(λ) − PG(u;k+1,l−1)(λ) = PG(u;k−l+1,1)(λ) − PG(u;k−l+2,0)(λ). Removal of the end vertex of the single attached path in G(u; k − l + 2, 0) gives

PG(u;k−l+2,0)(λ) = λPG(u;k−l+1,0)(λ) − PG(u;k−l,0)(λ), 3.5. The Characteristic Polynomial Approach 107 while the removal of the leaf adjacent to u in G(k − l + 1, 1) gives

PG(u;k−l+1,1)(λ) = λPG(u;k−l+1,0)(λ) − PG(u;k−l+1,0)−u(λ).

Therefore, PG(u;k,l)(λ) − PG(u;k+1,l−1)(λ) = PG(u;k−l,0)(λ) − PG(u;k−l+1,0)−u(λ). A crucial observation here is that the graph G(u; k − l + 1, 0) − u is isomorphic to a proper spanning subgraph of G(u; k−l, 0), which by the following Lemma 3.5.5 implies that PG(u;k,l)(λ) − PG(u;k+1,l−1)(λ) < 0 for all x ≥ λG(u;k,l),1 > λG(u;k−l,0),1, and hence that λG(u;k,l),1 > λG(u;k+1,l−1),1. Lemma 3.5.5 (Li–Feng [38]). If H is a proper spanning subgraph of a graph G, then (∀λ ≥ λG,1) PG(λ) ≤ PH (λ), (3.24) with strict inequality if G is connected. Proof. Let V be the vertex set of G. Diﬀerentiation of det(λI − A) provides the relation 0 X PG(λ) = PG−v(λ). (3.25) v∈V The lemma is now easily proved by induction on the number of vertices n = |V |. ∼ ∼ 2 2 If n = 2 then G = K2 and H = 2K1, so that PG(λ) = λ − 1 < λ = PH (λ) for all λ. Otherwise, if the lemma has already been proved for graphs with n − 1 vertices, then from (3.25) we have

0 0 X PG(λ) − PH (λ) = (PG−v(λ) − PH−v(λ)) ≤ 0 v∈V for all λ ≥ λG,1, as H − v is a proper spanning subgraph of G − v and λG,1 ≥ λG−v,1. Hence, PG(λ) − PH (λ) is decreasing for λ ≥ λG,1. Since for λ = λG,1 we have PG(λG,1) = 0 and PH (λG,1) ≥ 0, because λH,1 ≤ λG,1, we conclude that PG(λ) ≤ PH (λ) for all λ ≥ λG,1. If G is connected, then λH,1 < λG,1 and PH (λG,1) > 0, which implies strict inequality in (3.24). We are now in a position to prove the second half of the classical result mentioned in the previous section.

Theorem 3.5.6. If T is a tree on n vertices, then λPn,1 ≤ λT,1 with equality if and ∼ only if T = Pn. Proof. If T is not a path, then it contains a vertex of degree at least 3. Among the vertices of degree ≥ 3, let u have the greatest eccentricity. Then, all other vertices of degree ≥ 3 are contained in a single component of T − u, which implies that at least two of these components are paths of lengths k and l that are attached at u in T , so that T =∼ T 0(k, l) with k ≥ l ≥ 1. Apply Lemma 3.5.4 repeatedly to obtain that λT,1 > λT 0(k+l,0),1. This shows that T cannot have the minimum spectral radius among trees. Moreover, the degree of the vertex u in T 0(k+l, 0) has 108 Chapter 3. Spectral Radius of Graphs

Figure 3.8: Graphs F1 and F2 from Lemma 3.5.7. decreased by one. Repeated application of the above procedure therefore decreases the sum of degrees of vertices having degree ≥ 3, indicating that it will eventually result in the path Pn as the tree with the minimum spectral radius. Difference of characteristic polynomials has been further employed in Han- sen–Stevanović[30] and in the series of papers [14, 36, 37] on graphs with minimum spectral radius and fixed diameter. We do not cover them here as all of them are relatively lengthy reductions of characteristic polynomials of appropriate graphs using Schwenk’s lemmas 3.5.1–3.5.3. Schwenk’s lemmas may yield equality of spectral radii by identifying common factors of characteristic polynomials of two similar graphs. An elegant example is contained in the following lemma. Lemma 3.5.7 (Lan–Li–Shi [36]). Let G and H be two connected graphs, and let u be a vertex of G and v a vertex of H. Let F1 and F2 be the graphs formed as shown in Fig. 3.8. Then, λF1,1 = λF2,1.

Proof. Using notation from Lan–Li–Shi [36], Lemma 3.5.1 applied to the cut edge st in F1 yields

PF1 (λ) = PG−•(λ)PG−•−•−H (λ) − PG(λ)PG−•(λ)PH (λ)

= PG−•(λ)[PG−•−•−H (λ) − PG(λ)PH (λ)] , while in F2 it yields

PF2 (λ) = PH−•(λ)PG−•−•−H (λ) − PG(λ)PH (λ)PH−•(λ)

= PH−•(λ)[PG−•−•−H (λ) − PG(λ)PH−•(λ)] .

Since G − • is a proper subgraph of F1, λG−•,1 < λF1,1, so λF1,1 is the largest root of PG−•−•−H (λ) − PG(λ)PH (λ). Similarly, H − • is a proper subgraph of

F2, so λF2,1 is the largest root of PG−•−•−H (λ) − PG(λ)PH (λ) as well. Hence, λF1,1 = λF2,1. 3.5. The Characteristic Polynomial Approach 109

If a graph has a suﬃciently symmetric structure, its simple eigenvalues can easily be factored out from the characteristic polynomial. Although we are mainly interested in the adjacency matrix in this chapter, we deﬁne the next concept in terms of general matrices, as it may equally well be applied to other matrices r associated with graphs. A partition π : V = ∪i=1Vi of the index set of the square matrix A is called an equitable partition if there exists an r × r matrix B such P ∈ { } ∈ that v∈Vj Au,v = Bi,j, for all i, j 1, . . . , r and all u Vi. Recall that B is the quotient matrix of A with respect to the partition π. A fundamental relation between the eigenvalues of A and B is described in the following lemma. Lemma 3.5.8 (Petersdorf–Sachs, [45]). If a real, symmetric matrix A has an equitable partition π of its index set, then the characteristic polynomial of the quotient matrix B corresponding to π divides the characteristic polynomial of A. Despite being stated in terms of characteristic polynomials, this lemma is most easily proved by resorting to eigenvalues and eigenvectors, assuming that the matrix A is diagonalizable (which is certainly the case for real, symmetric matrices). Proof. Let ρ be an eigenvalue of B with the corresponding eigenvector x. Form a vector y indexed by V such that yu = xi if u ∈ Vi. Then, for u ∈ Vi,

r r X X X X (Ay)u = Au,vyv = Au,vxj = Bi,jxj = ρxi = ρyu,

v∈V j=1 v∈Vj j=1 showing that y is an eigenvector of A corresponding to the eigenvalue ρ. If ρ has multiplicity k as the eigenvalue of B, then there exists a set of k linearly independent eigenvectors of B corresponding to ρ. Linear independence is obviously preserved by the above construction of the eigenvectors of A, so that ρ then has multiplicity at least k as the eigenvalue of A. Consequently, the characteristic polynomial of B divides the characteristic polynomial of A.

r In the setting of adjacency matrices of graphs, the partition π : V = ∪i=1Vi of the vertex set V of the graph G is an equitable partition if each vertex of Vi is adjacent to the same number bij of vertices of Vj for each i, j ∈ {1, . . . , r}. The graph H whose adjacency matrix is the quotient matrix B = (bij) is then usually called the divisor of G. By the previous lemma the spectral radius of the divisor H is an eigenvalue of G, but the lemma does not state whether the spectral radius of G has to be an eigenvalue of H as well. This is nevertheless true, but we will postpone its proof to the next section, as it depends on counting walks in both G and H. Divisors are often used when an automorphism φ of a graph G is available, since the orbits of φ form a natural equitable partition of the vertex set of G.

Example 3.5.9 (Belardo–Li-Marzi–Simi´c[3]). Let T = DSp,q be a double star Sp+1(u − v)Sq+1, where u is the center of Sp+1 and v is the center of Sq+1. T has 110 Chapter 3. Spectral Radius of Graphs

an automorphism whose four orbits are formed by: all the leaves of Sp+1, the vertex u, the vertex v, and all the leaves of Sq+1. The quotient matrix corresponding to this equitable partition is  0 1 0 0   p 0 1 0    .  0 1 0 q  0 0 1 0 Its characteristic polynomial is λ4 − (p + q + 1)λ2 + pq, and the spectral radius of T is equal to the largest root of this polynomial:

s p p + q + 1 + (p + q + 1)2 − 4pq λT,1 = . 2 The concept of divisors is well established in both graph theory and matrix theory. A good overview of further development of the divisor concept may be found in Cvetkovi´c–Doob–Sachs [18, Sect. 4.6].

3.6 Walk Counting

The third approach used to estimate spectral radii of graphs stems directly from Theorem 3.2.1, which relates the number of walks in a graph to the powers of its adjacency matrix. We have seen in Section 3.2 that the spectral decomposition of A implies the identities (3.10) for Nk the number of all walks of length k, and (3.11) for Mk the number of all walks of length k. Before proceeding with the use of these expressions in comparisons of spectral radii, let us ﬁrst ﬁnish the proof that the spectral radius λG,1 necessarily appears as an eigenvalue of any divisor H of G. Denote by µ1 > ··· > µs all distinct eigenvalues of G. For i = 1, . . . , s let mi denote the multiplicity of the eigenvalue i i µi, and let x1, . . . , xmi denote the orthonormal eigenvectors corresponding to µi. Further, let mi X X i 2 Ci = xj,u , (3.26) j=1 u∈V so that (3.10) may be rewritten as

s X k Nk = Ciµi . (3.27) i=1

The eigenvalue µi with Ci =6 0 is called the main eigenvalue of G, as it has an inﬂuence on the number of walks. We see from (3.26) that Ci = 0 happens exactly when all the eigenvectors of µi are orthogonal to the all-ones vector 1. For example, if G is an r-regular graph, then 1 is its principal eigenvector, so the eigenvectors of all other eigenvalues are orthogonal to it. Hence, the degree r is the only main 3.6. Walk Counting 111 eigenvalue of an r-regular graph. As a matter of fact, the spectral radius of G is always its main eigenvalue, as the principal eigenvector is positive and cannot be orthogonal to 1. Sr Let π : V = i=1 Vi be an equitable partition such that each vertex of Vi is adjacent to the same number bij of vertices of Vj for each i, j ∈ {1, . . . , r}. The divisor H with the adjacency matrix B = (bij) is then a directed multigraph on the vertex set {1, . . . , r}, as the matrix B need not be symmetric and may have entries greater than 1. However, what is important is that the numbers of walks in G and H are well related. First, note that in specifying a walk in a multigraph we have to list its edges in addition to its vertices, as several edges may exist between any given pair of vertices, but other than that equations (3.10) and (3.11) hold for multigraphs as well. Let us now create an adjacency list Lu,j for each vertex u of G and each j ∈ {1, . . . , r}: if u ∈ Vi then Lu,j contains an ordered list of bij 0 neighbors of u in Vj. Similarly, let Lij, i, j ∈ {1, . . . , r}, represent the ordered list of bij edges connecting the vertices i and j in H. We may now establish a mapping between walks in G and in H by mapping adjacent vertices in G to incident edges in H. For a walk W : v0, . . . , vk of G containing the edges v0v1, . . . , vk−1vk, the 0 corresponding walk W of H is obtained in the following way: the vertex v0 of G is mapped to the index p of H such that v0 ∈ Vp, while for i = 1, . . . , k, if vi ∈ Vq is the a-th entry in the adjacency list Lvi−1,q, then the edge vi−1vi of W is mapped 0 to the a-th edge in the adjacency list Li−1,q, while the vertex vi is mapped to the index q. This way, if we know the associated walk W 0 in H and the starting vertex v0 in G, we can fully reconstruct the original walk W . This shows that the Pr k number of walks of length k in G may be given as NG,k = p,q=1 |Vp|(B )pq. Let 0 ν1 > ··· > νt be the distinct eigenvalues of B, and for each eigenvalue νj let mj j j be its multiplicity and y1, . . . , y 0 be the orthonormal eigenvectors corresponding mj to νj. From the spectral decomposition of B we have

0 0 r t mj r r mj r X X k X j j X k X X j j X k NG,k = |Vp| νj yi,pyi,q = νj |Vp| yi,pyi,q = Djνj , p,q=1 j=1 i=1 j=1 p,q=1 i=1 j=1 (3.28) m0 Pr P j j j where Dj = p,q=1 |Vp| i=1 yi,pyi,q. From (3.27) and (3.28) we now have that Pt k Pr k the equality i=1 Ciµi = j=1 Djνj holds for each integer k ≥ 0. Since the infinite power vectors (1, z, z2,...) are linearly independent for distinct values of z, this proves that the set of eigenvalues {µi | Ci =6 0} must be equal to the set {νj | Dj}, and moreover the corresponding C and D coefficients must be equal. Hence, we have proved Theorem 3.6.1 (Cvetković[17]). The main eigenvalues of a graph G appear as the eigenvalues of any of its divisors. We can now turn our focus on using the expressions (3.10) and (3.11) to compare spectral radii of graphs. Note that usually we do not aim to find explicit 112 Chapter 3. Spectral Radius of Graphs expressions for the numbers of walks, as these are too complicated for us to be able to work with them efficiently (except in very rare cases). Take a look, for example, at the expression for the number of walks of length k in the path Pn:

b n+1 c k+1 2 2 X 2 2l − 1 π k 2l − 1 NP ,k = cot cos π ; n n + 1 n + 1 2 n + 1 l=1 see Collatz–Sinogowitz[15]. Instead, we will mostly aim to compare the numbers of walks in two graphs and show that one of them has fewer walks of any given length than the other. Our starting point in this direction is the following lemma that relates the spectral radius to the numbers of walks. √ k Lemma 3.6.2 (Stevanovi´c,[√ 52]). For a connected graph G, λG,1 = limk→∞ Nk 2k and λG,1 = limk→∞ M2k.

Proof. Bipartite graphs have to be treated separately here, due to the fact that their spectrum is symmetric with respect to zero; see Cvetkovi´c[16]. In such cases, 0 00 0 00 λn = −λ1 is also a simple eigenvalue of G, and if V = V ∪ V , V ∩ V = ∅, represents a bipartition of the vertex set of G, then the eigenvector corresponding to λn satisﬁes ( 0 x1,u, if u ∈ V , xn,u = 00 −x1,u, if u ∈ V .

Therefore, v u 2 2 n−1 2k+1 2 2k+1p 2k+1u X X X λi X N2k+1 = λ1 t x1,u − xn,u + xi,u λ1 u∈V u∈V i=2 u∈V v u n−1 2k+1 2 2k+1u X X X λi X = λ1 t2 x1,u x1,u + xi,u , λ1 u∈V 0 u∈V 00 i=2 u∈V

v u 2 2 n−1 2k 2 2pk 2uk X X X λi X N2k = λ1 t x1,u + xn,u + xi,u λ1 u∈V u∈V i=2 u∈V v u 2 2 n−1 2k 2 2uk X X X λi X = λ1 t2 x1,u + 2 x1,u + xi,u . λ1 u∈V 0 u∈V 00 i=2 u∈V √ k P P P 2 Then, limk→∞ Nk =λ1 since both ( u∈V 0 x1,u)( u∈V 00 x1,u) and ( u∈V 0 x1,u) P 2 + ( u∈V 00 x1,u) are positive constants and, for each i = 2, . . . , n − 1, we have P 2 |λi/λ1| < 1, with u∈V xi,u being another constant. 3.6. Walk Counting 113

On the other hand, if G is not bipartite, then −λ1 < λn, so that v u 2 n k 2 pk uk X X λi X Nk = λ1 t x1,u + xi,u . λ1 u∈V i=2 u∈V √ k P 2 The equality limk→∞ Nk = λ1 then follows from the fact that u∈V x1,u is P 2 a positive constant, with |λi/λ1| < 1 and u∈V xi,u being a constant for each i = 2, . . . , n. The proof for the limit of closed walks follows along the same lines. For closed walks we are only taking even walks into account, as there are no odd walks when the graph is bipartite. The previous lemma therefore enables the implications

(∀k ≥ 0) NG,k ≥ NH,k =⇒ λG,1 ≥ λH,1,

(∀k ≥ 0) MG,2k ≥ MH,2k =⇒ λG,1 ≥ λH,1, for connected graphs G and H. As a matter of fact, inequalities on the left-hand side need not be satisfied for all walk lengths, it is enough to have them just for infinitely many different lenghts. Lemma 3.6.3 (Stevanović,[52]). Let G and H be connected graphs such that, for an infinite sequence of indices k0 < k1 < ··· , we have either

(∀i ≥ 0) NG,ki ≥ NH,ki (3.29) or

(∀i ≥ 0) MG,2ki ≥ MH,ki . (3.30)

Then, λG,1 ≥ λH,1. Proof. We focus here on the numbers of all walks. The proof for the closed walks goes along the same lines. From Lemma 3.6.2 we get s λ N lim G,1 k G,k = 1, k→∞ λH,1 NH,k which implies s λG,1 k NG,k (∀ε > 0)(∃k0)(∀k ≥ k0) > (1 − ε) . λH,1 NH,k

The condition (3.29), with i0 taken to be the smallest index such that ki0 ≥ k0, now implies λG,1 (∀ε > 0)(∃i0)(∀i ≥ i0) > 1 − ε. λH,1 114 Chapter 3. Spectral Radius of Graphs

Since λG,1 and λH,1 do not depend on i, the previous expression actually means

λ (∀ε > 0) G,1 > 1 − ε, λH,1 which is equivalent to λG,1 ≥ λH,1.

In order for the previous lemma to imply λG,1 > λH,1 instead of (3.29) or (3.30), one would need to provide the much harder estimate

ε ki (∃ε > 0)(∀i0)(∃i ≥ i0) NG,k ≥ 1 + NH,k . i 1 − ε i

As we will see from later examples, it already takes quite a lot of work to prove (3.29) or (3.30), so in this section we will be satisfied with the weak inequality alone. The inability of Lemma 3.6.3 to imply strict inequality means that it cannot be used to characterize graphs with the maximum spectral radius in a certain class, although it can be used to characterize the value of this spectral radius and pinpoint an example of such an extremal graph (which often happens to be unique). The following graph composition enables one to create larger graphs whose numbers of walks will monotonically depend on the numbers of walks in their constituents, and as such enable the use of the walk-limit Lemmas 3.6.2 and 3.6.3. Definition 3.6.4. Let F and G be the graphs with disjoint vertex sets V (F ) and V (G). For p ∈ N, let u1, . . . , up be distinct vertices from V (F ), and let v1, . . . , vp be distinct vertices from V (G). Assume, in addition, that there is no pair (i, j), i =6 j, such that both uiuj is an edge of F and vivj is an edge of G. The multiple coalescence of F and G with respect to the vertex lists u1, . . . , up and v1, . . . , vp, denoted by F (u1 = v1, . . . , up = vp)G, is the graph obtained from the union of F and G by identifying the vertices ui and vi for each i = 1, . . . , p. The multiple coalescence is a generalization of the standard coalescence of two vertex-disjoint graphs, which is obtained by identifying a single pair of vertices, one from each graph; see Cvetković–Doob–Sachs [18]. Fig. 3.9 shows an example of multiple coalescence of the graphs F and G, with respect to the selected vertices u1, u2, u3 and v1, v2, v3. The above assumption that for any i =6 j it is not the case both that uiuj is an edge of F and vivj is an edge of G serves to prevent the creation of multiple edges in the multiple coalescence. This assumption is needed later, as our goal will be to have each walk in the multiple coalescence clearly separated into smaller parts all of whose edges belong to only one of its constituents. In such a setting, the vertices v1, . . . , vp may be considered as the entrance points for a walk coming from F to enter G (and vice versa). Our main tool is the following general theorem, whose slightly weaker versions appear in Stevanović[52] and Huang–Li–Wang [34]. 3.6. Walk Counting 115

Figure 3.9: An example of multiple coalescence of two graphs, [52].

Theorem 3.6.5. Let p be an arbitrary natural number. Let F,F 0,G, and G0 be four graphs with at least p vertices each, such that {F,F 0}×{G, G0} are four pairs of vertex-disjoint graphs. Choose four p-tuples of distinct vertices (u1, . . . , up) ∈ V (F ), 0 0 0 0 0 0 (u1, . . . , up) ∈ V (F ), (v1, . . . , vp) ∈ V (G), and (v1, . . . , vp) ∈ V (G ). Let H and 0 0 0 0 H be the multiple coalescences H = F (u1 = v1, . . . , up = vp)G and H = F (u1 = 0 0 0 0 0 v1, . . . , up = vp)G , such that both H and H are connected. Let AF ,AF 0 ,AG, 0 0 and AG0 be the adjacency matrices of F,F ,G, and G , respectively. If for each 1 ≤ i, j ≤ p (including the case i = j) and for each k ≥ 1 the conditions

MF,2k ≤ MF 0,2k, (3.31) k k ≤ 0 0 0 (AF )ui,uj (AF )ui,uj , (3.32)

MG,2k ≤ MG0,2k, (3.33) k k ≤ 0 0 0 (AG)vi,vj (AG )vi,vj (3.34) all hold, then λH,1 ≤ λH0,1. Proof. Let us count closed walks of length 2k in H. From the fact that F and G, as constituents of H, do not have common edges, we see that the number of closed walks in H, all of whose edges belong to the same constituent, is equal to MF,2k + MG,2k. The remaining closed walks in H contain edges from both F and G. Any such closed walk W can be decomposed into a sequence of subwalks W : W0,W1,..., W2l−1, for some l ∈ N, such that the edges of the even-indexed subwalks W0,..., W2l−2 all belong to F , while the edges of the odd-indexed subwalks W1,...,W2l−1 all belong to G. As a walk can enter G from F only through one of the entrance points, we see that the end vertices of the even-indexed subwalks be- 116 Chapter 3. Spectral Radius of Graphs

long to {u1, . . . , up}, while the end vertices of the odd-indexed subwalks belong to {v1, . . . , vp}. Let (i0, . . . , i2l−1) denote the 2l-tuple of indices such that, for j = 0, . . . , l − 1, the walk W2j goes from ui2j to ui2j+1 (= vi2j+1 ) in F , while the walk W2j+1 goes from vi2j+1 to vi2j+2 (= ui2j+2 ) in G. (The addition is modulo 2l, so that i2l = i0.) Furthermore, let kj denote the length of the walk Wj for j = 0,..., 2l − 1. The 4l-tuple (i0, . . . , i2l−1; k0, . . . , k2l−1) will be called the signature of the closed walk W . Due to the fact that the walk W is closed, its signatures are rotationally equivalent in the sense that the above signature is identical to the signature

(i2p, . . . , i2l−1, i0, . . . , i2p−1; k2p, . . . , k2l−1, k0, . . . , k2p−1) for each p = 1, . . . , l − 1. In order to assign a unique signature to W , we may assume its signature is chosen as lexicographically minimal among all rotationally equivalent signatures. For any feasible signature (i0, . . . , i2l−1; k0, . . . , k2l−1), the number of closed walks in H with this signature is equal to

l−1 l−1 Y ki Y ki 2k (A 2j ) (A 2j+1 ) , F ui2j ,ui2j+1 G vi2j+1 ,vi2j+2 j=0 j=0 as the ﬁrst vertex may be chosen arbitrarily among the vertices on the closed walk. The argument is identical for closed walks of length 2k in H0: the number of closed walks, all of whose edges belong to the same constituent of H, is equal to MF 0,2k + MG0,2k, while the number of closed walks with the feasible signature (i0, . . . , i2l−1; k0, . . . , k2l−1) is equal to

l−1 l−1 Y ki2j Y ki2j+1 2k (A 0 )u0 ,u0 (A 0 )v0 ,v0 . F i2j i2j+1 G i2j+1 i2j+2 j=0 j=0

From conditions (3.31)–(3.34) we see that, for any feasible signature (i0,..., i2l−1; k0, . . . , k2l−1), the number of closed walks with this signature in H is less than or equal to the number of closed walks with this signature in H0. Summing over all feasible signatures we get that MH,2k ≤ MH0,2k for each k ≥ 1 and, from Lemma 3.6.3, we finally conclude that λH,1 ≤ λH0,1. Although the previous theorem is defined in a very general way, its elements may coincide in order to simplify proving conditions (3.31)–(3.34). Note first that 0 it does not ask F and F to be vertex-disjoint. Also, while the vertices u1, . . . , up have to be distinct from each other, they may coincide with some (or all) of the 0 0 0 vertices u1, . . . , up. Similarly, G and G need not be vertex disjoint, and the p- 0 0 tuples (v1, . . . , vp) and (v1, . . . , vp) may have vertices in common. As a matter of 0 0 0 fact, in practice, we will most often have F = F ,(u1, . . . , up) = (u1, . . . , up) and 0 0 0 G = G , with only (v1, . . . , vp) and (v1, . . . , vp) representing two different p-tuples 3.6. Walk Counting 117

Figure 3.10: Example of multiple coalescences with λH,1 ≤ λH0,1. of the vertices of G. In such a case, conditions (3.31)–(3.33) are automatically satisﬁed, and we need to prove only condition (3.34). An example of a corollary of Theorem 3.6.5 is found in Du–Liu [22] and several references cited there. Corollary 3.6.6 (Du–Liu [22]). Let u and v be two vertices of the graph G = (V,E) with adjacency matrix A. Further, let w1, . . . , wr be the vertices of G such that uwi, vwi ∈/ E. Let Gu = G + uw1 + ··· + uwr and Gv = G + vw1 + ··· + vwr. If k k k k (A )u,u ≤ (A )v,v and (A )u,wi ≤ (A )v,wi for each k ≥ 0 and each i = 1, . . . , r, then MGu,k ≤ MGv ,k.

Proof. Let Sr+1 be a star with the center vertex s and the leaves t1, . . . , tr. Then Gu and Gv are the multiple coalescences Gu = Sr+1(s = u, t1 = w1, . . . , tr = wr)G and Gv = Sr+1(s = v, t1 = w1, . . . , tr = wr)G. Conditions (3.31)–(3.33) are then automatically satisfied, as is condition (3.34) for pairs of vertices wi, wj. For the k k k k remaining part of condition (3.34): (A )u,u ≤ (A )v,v and (A )u,wi ≤ (A )v,wi is provided as an assumption, so that Theorem 3.6.5 now yields MGu,k ≤ MGv ,k. In the sequel, we showcase a few results proving (3.34) in specific cases. Lemma 3.6.7. Let G be a connected graph with adjacency matrix A. If u is a leaf k k of G and v is its unique neighbor then, for each k, (A )u,u ≤ (A )v,v. k Proof. The term (A )u,u counts closed walks of length k starting and ending at u. Since v is the only neighbor of u, each such walk necessarily has the form W : uv, W ∗, vu, where W ∗ is a closed walk of length k−2 starting and ending at v. If we denote by f(W ) the closed walk f(W ): vu, uv, W ∗, then it is straightforward to see that the function f injectively maps closed walks of length k that start at k k u into closed walks of length k that start at v. Hence (A )u,u ≤ (A )v,v. Using the previous lemma as the “feeder” result within Theorem 3.6.5 implies 0 that λH,1 ≤ λH0,1 holds for the graphs H and H illustrated in Fig. 3.10. It seems that a lemma equivalent to this one first appeared in Du–Zhou [23], where it was used in characterization of connected graphs with maximum Estrada index and given number of cut edges. As a matter of fact, most current results on walk counting appeared in the literature on the Estrada index, which is defined in 118 Chapter 3. Spectral Radius of Graphs

Pn λi P Estrada [24] as EE = i=1 e = k≥0 Mk/k!. The Estrada index is, similarly to the spectral radius, monotonically dependent on the numbers of closed walks, so that the same walk-counting results apply to both of these invariants. The difference is the compatibility of EE with strict inequality among closed walk counts, which is absent from the spectral radius, due to the appearance of the limit in Lemma 3.6.2. Before stating the next lemma, let us first introduce a few definitions. For a graph G and its vertices u, v, and w, let NG,k,u,v denote the number of walks of length k from u to v and let NG,k,u,v,[w] denote the number of walks of length k from u to v that pass through the vertex w. Similarly, let MG,k,u denote the number of closed walks of length k starting at u and MG,k,u,[w] the number of such walks passing through w.

Lemma 3.6.8 (Du–Zhou, [23]). Let G be a connected graph and Ps a path on 0 s ≥ 2 vertices. Let u be a vertex of G and v, v the two end vertices of Ps. Then, 0 MG(u=v)Ps,k,v ≥ MG(u=v)Ps,k,v . 0 Proof. Denote the vertices along the path Ps as v = v1, v2, . . . , vs−1, vs = v . Note that in the coalescence H = G(u = v1)Ps we have

MH,k,vs = MPs−v1,k,vs + MH,k,vs,[v1],

MH,k,v1 = MG(u=v1)Ps−vs,k,v1 + MH,k,v1,[vs].

The map vl → vs−l+1, l = 1, . . . , s, embeds the path Ps −v1 into G(u = v1)Ps −vs such that vs is mapped to v1, showing that MPs−v1,k,vs ≤ MG(u=v1)Ps−vs,k,v1 . Further, each closed walk W in H starting at vs and passing through v1 may be uniquely decomposed as the sequence of two walks W1,W2, where W1 is the shortest initial subwalk of W from vs to v1 consisting of a vs-v2 walk in Ps − v1 and a single edge v2v1, while W2 is the remaining v1-vs subwalk of W . Then, P 0 0 MH,k,vs,[v1] = k0≤k NPs−v1,k −1,vs,v2 NH,k−k ,v1,vs . Similarly, X 0 0 MH,k,v1,[vs] = NG(u=v1)Ps−vs,k −1,v1,vs−1 NH,k−k ,vs,v1 . k0≤k

The same embedding vl → vs−l+1, l = 1, . . . , s, shows that

0 0 NPs−v1,k −1,vs,v2 ≤ NG(u=v1)Ps−vs,k −1,v1,vs−1 , which implies MH,k,vs,[v1] ≤ MH,k,v1,[vs] and, hence, MH,k,vs ≤ MH,k,v1 . The next result, which orders the numbers of closed walks starting at diﬀerent vertices of a path, will enable us to obtain yet another proof of the Li–Feng Lemma 3.5.4.

Theorem 3.6.9 (Ili´c–Stevanovi´c,[35]). Let Pn be a path on vertices 1, . . . , n. Then, for every k ≥ 0 and 1 ≤ s ≤ n − 1,

n−s+1 n+s−1 MPn,k,1,s ≤ MPn,k,2,s+1 ≤ ... ≤ M . (3.35) Pn,k,d 2 e,d 2 e 3.6. Walk Counting 119

Proof. Let  0 1 0 ... 0 0 0   1 0 1 ... 0 0 0     0 1 0 ... 0 0 0     ......  A =  ......     0 0 0 ... 0 1 0     0 0 0 ... 1 0 1  0 0 0 ... 0 1 0 be the adjacency matrix of the path Pn. The theorem will follow from the inequalities k k (A )i−1,j−1 ≤ (A )i,j (3.36) which we prove, by induction on k, to hold for all 2 ≤ i, j ≤ n such that i+j ≤ n+1. First, each diagonal of A0 = I and A1 = A is either all-zeroes or all-ones, proving the basis of induction for k = 0 and k = 1. Suppose now that (3.36) has been proved for some k ≥ 1. The expression Ak+1 = Ak · A yields k+1 k k (A )i−1,j−1 = (A )i−1,j−2 + (A )i−1,j, k+1 k k (A )i,j = (A )i,j−1 + (A )i,j+1.

k k (We assume (A )i−1,0 = 0 and (A )i,n+1 = 0 above to avoid dealing separately k k with the endpoints of Pn.) Then, (A )i−1,j−2 ≤ (A )i,j−1 holds by the inductive k k hypothesis and the nonnegativity of (A )i,j−1. If i+j +1 ≤ n+1, then (A )i−1,j ≤ k (A )i,j+1 also holds by the inductive hypothesis. For i + j + 1 = n + 2, from the k automorphism φ: i → n + 1 − i of Pn and the symmetry of A we have k k k (A )i−1,j = (A )n+1−j,n+2−i = (A )i,j+1. This proves (3.36) for k + 1 as well. Due to the automorphism φ: i → n + 1 − i, i = 1, . . . , n, the previous result means that each diagonal of Ak parallel to the main diagonal is unimodal with peak at its middle entry. Let us recall that the Li–Feng Lemma 3.5.4 claims that if p ≥ q ≥ 1, then

λG(u;p,q),1 > λG(u;p+1,q−1),1, (3.37) where G(u; p, q) denotes the graph obtained from a nontrivial graph G by attaching pendant paths of lengths p and q at the vertex u of G. Note that G(u; p, q) can be viewed as coalescence G(u = q + 1)Pp+q+1, where the pendant paths of lengths p and q form a single path Pp+q+1, whose vertices are enumerated starting from the outer end vertex of the shorter path Pq+1 toward u and continuing toward the outer end vertex of the longer path Pp+1. Replacing the strict inequality > in (3.37) with the weak inequality ≥,(3.37) now becomes a corollary of Theorem 3.6.5 and Theorem 3.6.9 with s = 1. Another lemma is attributed to Li and Feng as well: 120 Chapter 3. Spectral Radius of Graphs

Figure 3.11: Graphs G(u, v; p, q) and G(u, v; p + 1, q − 1) from Lemma 3.6.10.

Lemma 3.6.10 (Li–Feng, [38]). Let u and v be two adjacent vertices of a connected graph G and, for positive integers p and q, let G(u, v; p, q) denote the graph obtained from G by attaching a pendant path of length p at u and a pendant path of length q at v. If p ≥ q ≥ 1, then λG(u,v;p,q),1 > λG(u,v;p+1,q−1),1. The proof of this lemma, again with > replaced with ≥, follows by observing that G(u, v; p, q) and G(u, v; p + 1, q − 1) are actually multiple coalescences: ∼ G(u, v; p, q) = G − uv(u = q + 2, v = q + 1)Pp+q+2, ∼ G(u, v; p + 1, q − 1) = G − uv(u = q + 1, v = q)Pp+q+2, with vertices of Pp+q+2 enumerated starting from the shorter path Pq+1, respec- ∗ tively Pq+2 in G (p + 1, q − 1); see Fig. 3.11. Theorem 3.6.5 now requires that, for k ≥ 1,

MPp+q+2,k,q+2 ≥ MPp+q+2,k,q+1,

MPp+q+2,k,q+1 ≥ MPp+q+2,k,q,

NPp+q+2,k,q+2,q+1 ≥ NPp+q+2,k,q+1,q, which are special cases of Theorem 3.6.9 with s = 1 and s = 2 (and additionally with MPp+q+2,k,q+2 = MPp+q+2,k,q+1 in the case p = q due to the automorphism of P2q+2). Theorem 3.6.9 is easily generalized to rooted products of graphs. Definition 3.6.11 (Godsil–McKay, [27]). Let H be a labeled graph on n vertices, and let G1,...,Gn be a sequence of n rooted graphs. The rooted product of H by G1,...,Gn, denoted as H[G1,...,Gn], is the graph obtained by identifying the root of Gi with the i-th vertex of H for i = 1, . . . , n. In the case when all the rooted graphs Gi, i = 1, . . . , n, are isomorphic to a rooted graph G, we denote H[G, .n .) ., G] simply as H[G, n]. Theorem 3.6.12 (Stevanović,[52]). Let n be a positive integer and let G be an arbitrary rooted graph. Denote by G1,...,Gn the copies of G, and for any vertex u of G, denote by ui the corresponding vertex in the copy Gi, i = 1, . . . , n. Then, 3.6. Walk Counting 121 for any two (not necessarily different) vertices u and v of G, every k ≥ 0 and every 1 ≤ s ≤ n − 1, the following holds:

MP [G,n],k,u ,v ≤ MP [G,n],k,u ,v ≤ · · · ≤ MP [G,n],k,u ,v . (3.38) n 1 s n 2 s+1 n d n−s+1 e d n+s−1 e 2 2 The proof of this theorem goes along the same lines as the proof of Theorem 3.6.9 after observing that the numbers of walks between the vertices of Pn[G, n] are strongly governed by the numbers of walks between the roots of the copies of G in Pn[G, n]. Let r denote the root of G, so that r1, . . . , rn then also denote the vertices of Pn in the rooted product Pn[G, n]. If s = 1, the number of k-walks between ui and vi whose edges fully belong to Gi is, obviously, equal to the number of k-walks between u and v in G. Otherwise, if 0 a k-walk W between ui and vi contains other edges of Pn[G, n], then let W denote 0 the longest subwalk of W such that W is a closed walk starting and ending at ri: 0 simply, the ﬁrst edge of W is the ﬁrst edge of W that does not belong to Gi, and 0 the last edge of W is the last edge of W that does not belong to Gi. It is then easy to see that the number of k-walks between ui and vi in Pn[G, n] is determined by the numbers of walks between u and v in G, and the numbers of closed walks (of lengths k and less) starting and ending at ri in Pn[G, n]. If s > 1, the number of k-walks between ui in the copy Gi and vs+i−1 in the copy Gs+i−1 is governed by the numbers of walks between u and r in G (that get mapped to the walks between ui and ri in Gi), the numbers of walks between ri and rs+i−1 in Pn[G, n], and the numbers of walks between r and v in G (that get mapped to the walks between ri+1 and vi+1 in Gi+1). In any case, an important consequence is that the chain of inequalities (3.38) follows from

MP [G,n],k,r ,r ≤ MP [G,n],k,r ,r ≤ · · · ≤ MP [G,n],k,r ,r . n 1 s n 2 s+1 n d n−s+1 e d n+s−1 e 2 2

These inequalities may be written brieﬂy as MPn[G,n],k,ri−1,rj−1 ≤ MPn[G,n],k,ri,rj , for 2 ≤ i, j ≤ n, i + j ≤ n + 1, which are then proved by induction on k, as in the proof of Theorem 3.6.9. Theorem 3.6.12 allows us to generalize the Li–Feng lemmas to rooted products of graphs. Lemma 3.6.13 (Stevanovi´c,[52]). Let G be a rooted graph, H a connected graph, and p and q two positive integers. For a vertex u of H, suppose that H contains a rooted subgraph G0, with u as its root, that is isomorphic to the rooted graph G. Let H(u, G; p, q) denote the graph obtained from H by identifying the rooted subgraph 0 G with the (q + 1)-st copy of G in the rooted product Pp+q+1[G, p + q + 1]; see Fig. 3.12). If p ≥ q ≥ 1, then λH(u,G;p,q),1 ≥ λH(u,G;p+1,q−1),1.

Lemma 3.6.14 (Stevanovi´c,[52]). Let G be a rooted graph, H a connected graph, and p and q two positive integers. For two adjacent vertices u and v of H, suppose that H contains two vertex-disjoint rooted subgraphs G0, with a root u, and G00, 122 Chapter 3. Spectral Radius of Graphs

Figure 3.12: Graphs H(u, G; p, q) and H(u, v, G; p, q). with a root v, both isomorphic to the rooted graph G. Let H(u, v, G; p, q) denote the graph obtained from H by identifying the rooted subgraph G0 with the (q + 2)-nd copy of G and the rooted subgraph G00 with the (q + 1)-st copy of G in the rooted product Pp+q+2[G, p + q + 2]; see Fig. 3.12. If p ≥ q ≥ 1, then λH(u,v,G;p,q),1 ≥ λH(u,v,G;p+1,q−1),1.

These generalized lemmas further allow us to prove the analogue of the classical result on the spectral radius of trees.

Theorem 3.6.15 (Stevanovi´c,[52]). Let G be an arbitrary rooted graph. If T is a tree on n vertices, then λPn[G,n],1 ≤ λT [G,n],1 ≤ λSn[G,n],1.

By slightly tweaking the previous proofs, it is shown in Stevanović[52] that Lemmas 3.6.13 and 3.6.14 continue to hold even if we remove two copies of G attached to the end vertices of pendant paths in H(u, G; p, q) and H(u, v, G; p, q) (but the end vertices of these pendant paths are not removed). This was previ- ously conjectured in Belardo–Li-Marzi–Simić [2] for the special case of the rooted product with copies of the star Sr. The small problem remains, however, that both this conjecture and the Li–Feng lemmas claim the strict inequality, while our walk counting approach is able to prove only the weak inequality. In addition to previous results on counting walks in various cases of multiple coalescences, there are two more results in the literature where walk counting has been used to estimate the spectral radius of graphs. The first of these is Hayes’s theorem which has been mentioned earlier, in Section 3.3.

Theorem 3.6.16 (Hayes, [31]). Let G be a simple graph with the maximum vertex degree ∆, whose edges can be oriented such that at most d ≤ ∆/2 edges point away p from any vertex of G. Then, λ1 ≤ 2 d(∆ − d). 3.6. Walk Counting 123

Proof. We will prove k N2k ≤ 2n [4d(∆ − d)] , (3.39) from which the bound on λ1 will follow from the limit Lemma 3.6.2. Orient the edges of G so that at most d edges point away from any vertex of G. When an edge uv points from u to v, we say that u is an in-neighbor of v and that v is an out-neighbor of u. Create for each vertex u of G the list Lu of its neighbors such that the out-neighbors of u appear among the first d places in the list. The bound (3.39) is obtained by accounting for cases when a vertex of a walk appears among the first d places of the previous vertex’s neighbor list. Let W : v0, . . . , v2k be an arbitrary walk of length 2k in G, and let FW denote the set of indices i such that vi+1 appears among the first d places in the list Lvi . For any given subset F ⊆ {0,..., 2k − 1} there exist at most nd|F |(∆ − d)2k−|F | walks W such that FW = F : in fact, the first vertex v0 of W can be chosen in n ways, while each succeeding vertex vi+1 can be chosen in either d or at most ∆−d ways, depending on whether i ∈ F . The number of walks W such that |FW | ≥ k can now be bounded as 2k X 2k ndj(∆ − d)2k−j < 22kndk(∆ − d)k, j j=k due to d ≤ ∆ − d. To deal with the remaining walks, observe that at least k edges in W have the same orientation. If they are forward edges oriented from vi to vi+1, then |FW | ≥ k, which is dealt with above. If they are backward edges oriented from vi+1 to vi, then they become forward edges in the reverse walk WR : v2k, . . . , v0, so that |FWR | ≥ k. As each walk is in one-to-one correspondence with its reverse walk, 2k k k we conclude that there are also at most 2 nd (∆ − d) walks W with |FW | < k, from which (3.39) follows. Andriantiana–Wagner [1] have characterized maximum numbers of walks among trees with a given degree sequence. Let TD denote the set of trees with degree sequence D. The following definition describes the structure of extremal trees. Definition 3.6.17 (Andriantiana–Wagner, [1]). Let F be a rooted forest where each component has k levels. The leveled degree sequence of F is the sequence D = (V1,...,Vk), where Vi denotes the nonincreasing sequence of degrees of the vertices of F at the level i for 1 ≤ i ≤ k. The level greedy forest G(D) with leveled degree sequence

D = ((i1,1, . . . , i1,k1 ), (i2,1, . . . , i2,k2 ),..., (in,1, . . . , in,kn )) is obtained by the following inductive algorithm:

(i) label the vertices of the ﬁrst level g1,1, . . . , g1,k1 and assign degrees to these vertices such that deg(g1,j) = i1,j for all j = 1, . . . , k1; 124 Chapter 3. Spectral Radius of Graphs

Figure 3.13: A greedy tree with the leveled degree sequence ((4), (4,4,3,3), (3,3,3,3,2,2,2,2,2,1), (1,. . . ,1)).

(ii) assume that the vertices of level h have been labeled gh,1, . . . , gh,kh and a degree has been assigned to each of them; then, for all 1 ≤ j ≤ kh, label the neighbors of gh,j at level h + 1 by

g Pj−1 , . . . , g Pj h+1,1+ m=1(ih,m−1) h+1, m=1(ih,m−1)

and assign degrees to the newly labeled vertices such that deg(gh+1,j) = ih+1,j for all 1 ≤ j ≤ kh+1. A connected level greedy forest is called a level greedy tree. Finally, a tree T is called a greedy tree if one can choose a root vertex such that T becomes a level greedy tree whose leveled degree sequence D satisﬁes min(ij,1, . . . , ij,kj ≥ max(ij+1,1, . . . , ij+1,kj+1 ) for all 1 ≤ j ≤ n − 1. Fig. 3.13 shows an example of a greedy tree.

Theorem 3.6.18 (Andriantiana–Wagner, [1]). Let D be a degree sequence of a tree. Then, for any tree T ∈ TD and k ≥ 0, MT,k ≤ MG(D),k. The inequality is strict for suﬃciently large even k if T and G(D) are not isomorphic. As a corollary, the greedy tree G(D) has the largest spectral radius among all trees with degree sequence D, which was proved earlier by Bıyıko˘glu–Leydold[4] using the eigenvector approach. For two nonincreasing sequences of nonnegative numbers B = (b1, . . . , bn) and D = (d1, . . . , dn) we say that D majorizes B, denoted as B 4 D if, for each Pk Pk 1 ≤ k ≤ n, i=1 bi ≤ i=1 di. Andriantiana and Wagner have also shown that the numbers of closed walks in greedy trees are monotone with respect to majorization of degree sequences. Theorem 3.6.19 (Andriantiana–Wagner, [1]). Let B and D be degree sequences of trees with the same number of vertices such that B 4 D. Then, for any integer k ≥ 0, MG(B),k ≤ MG(D),k. The inequality is strict if B =6 D and k ≥ 4 is even. 3.6. Walk Counting 125

The n-vertex degree sequence D∆ = (∆,..., ∆, r, 1,..., 1) for some 1 ≤ r < ∆ majorizes all possible degree sequences of trees with maximum degree ∆, so that two previous theorems imply the conjecture from Ili´c–Stevanovi´c[35] that the greedy tree G(D∆) has the maximum number of closed walks among all trees with maximum degree ∆. Although unrelated to spectral radius, we would like to mention that Min- chenko–Wanless [39] have described a general procedure for expressing the numbers of closed walks in a regular graph in terms of its subgraph counts, which they then used to characterize connected 4-regular bipartite integral Cayley graphs, in Minchenko–Wanless [40]. Bibliography

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The Group Inverse of the Laplacian Matrix of a Graph by Stephen Kirkland

4.1 Introduction

What follows is a short, selective tour of some of the connections between weighted graphs and the group inverses of their associated Laplacian matrices. The presen- tation below draws heavily from Kirkland–Neumann [11, Ch. 7], and the interested reader can ﬁnd further results on the topic in that book. We note that Molitierno [13] also covers some of the material presented in this chapter, and so serves as another source for readers interested in pursuing this subject further. We have assumed that the reader has a working knowledge of matrix theory and knows the basics of graph theory. For background material on these topics, see Horn–Johnson [9] and Bondy–Murty [3], respectively. References to specialized results are given as needed, and a notation table follows for the reader’s convenience:

k 1k the all ones vector in R (the subscript is suppressed when the order is clear from the context); 0 a zero matrix or vector (the order is made clear from the context); α(G) the algebraic connectivity of the weighted graph G; diag(x) for a vector x ∈ Rn, diag(x) is the n × n diagonal matrix whose j-th diagonal entry is xj, j = 1, . . . , n; n ej the j-th standard unit basis vector in R (where n is clear from the context); 132 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

G \ e the weighted graph formed from G by deleting the edge e; G ∪ H the union of the vertex-disjoint weighted graphs G and H; G ∨ H the join of the vertex-disjoint unweighted graphs G and H; i ∼ j the edge between vertices i and j; Ik the identity matrix of order k (the subscript is suppressed when the order of the matrix is clear from the context); Jk,l the k × l all-ones matrix (if k =l, this is shortened to Jk, and sub- scripts are suppressed when the order is clear from the context); K(G) the Kirchhoﬀ index for the weighted graph G; K(G) the minimum Kirchhoﬀ index for the graph G; L{i}, L{i,j} the submatrix of L formed by deleting the i-th row and column, and the submatrix of L formed by deleting the i-th and j-th row and column, respectively; M > transpose of the matrix M; a similar notation applies to vectors; ||M||2 the spectral norm of the square matrix M (i.e., its largest singular value); R, R+ the real numbers, and the positive real numbers, respectively; r(i, j) the resistance distance between vertices i and j; ||x||1 the `1-norm of the vector x.

4.2 The Laplacian Matrix

In this section we introduce the Laplacian matrix of a weighted graph and outline some of its basic properties. Let G be an undirected graph with edge set E and vertices labelled 1, . . . , n. Suppose also that for each edge e ∈ E there is an associated weight, i.e., a positive number w(e). We think of the graph G, along with the associated weight function w, as a weighted graph; when each edge weight is 1 we say that G is unweighted. When vertices i and j are adjacent in G, we use the notation i ∼ j to denote the edge between them.

We construct the n × n matrix A = [ai,j]i,j=1,...,n from the weighted graph as follows:

( w(e), if i is adjacent to j and e is the edge between them ai,j = 0, if not.

Set D = diag(A1), where 1 is the all-ones vector in Rn. The Laplacian matrix for the weighted graph G is L = D − A. Throughout, we will typically suppress the explicit dependence on the weight function w, and simply refer to “the weighted graph G”. 4.2. The Laplacian Matrix 133

1u 2 u JJ J J uJ u u 3 5 4

Figure 4.1: Graph for Example 4.2.1

Example 4.2.1. Consider the graph in Fig. 4.1, and suppose that the graph is unweighted. We have the following as its Laplacian matrix:

 2 −1 0 0 −1   −1 2 0 0 −1     0 0 1 0 −1  .    0 0 0 1 −1  −1 −1 −1 −1 4

The next result records some of the basic properties of the Laplacian matrix for a weighted graph.

Proposition 4.2.2. Let G be a weighted graph on n vertices with edge set E and Laplacian matrix L. Then,

(i) For i, j = 1, . . . , n, we have  −w(e), if i is adjacent to j, e = i ∼ j,  li,j = 0, if i =6 j and i, j are not adjacent, P e∈E, e incident with j w(e), if i = j;

P > (ii) L = e∈E,e=i∼j w(e)(ei − ej)(ei − ej) ; (iii) impose any orientation on G (i.e., replace edges i ∼ j by directed arcs i → j). Construct the corresponding oriented incidence matrix H as follows: rows of H are indexed by vertices of G, columns of H are indexed by edges of G, and p for each e ∈ E, the column of H corresponding to e is w(e)(ei − ej), where i → j is the directed arc arising from the orientation of the edge e; then, L = HH>; (iv) L is positive semideﬁnite; (v) L is singular, with 1 as a null vector; (vi) if G is connected then the null space of L is spanned by 1; and 134 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

1u- 2 u J]J J J u-J u u 3 5 4

Figure 4.2: An orientation of the graph in Fig. 4.1

(vii) if G has k ≥ 2 connected components, say G1,...,Gk, then the nullity of L is k. Further, the null space of L is spanned by the vectors z(Gj), j = 1, . . . , k, where ( 1 if p ∈ Gj z(Gj)p = 0 if not.

Proof. (i) and (ii) are easily veriﬁed, while (iii) follows readily from (ii). For (iv), observe that from (iii) we have that for any x ∈ Rn, x>Lx = x>HH>x = > 2 ||H x||2 ≥ 0. (v) is immediate from (i). To show (vi), suppose that x is a null > vector for L. Then H x = 0, from which we deduce that xi = xj whenever i is adjacent to j in G. It follows that for any pair of vertices p, q, we must have xp = xq, since there is a path from p to q in G, say with p = p0, q = pk, and edges pj−1 ∼ pj, j = 1, . . . , k. Hence, xp0 = xp1 = ··· = xpk so that x is a scalar multiple of 1. Finally, (vii) follows from the fact that L can be written as a direct sum of the Laplacian matrices for G1,...,Gk. Example 4.2.3. Here we revisit the unweighted graph in Fig. 4.1. Consider the orientation of that graph depicted in Fig. 4.2. This then yields the following matrix as our oriented incidence matrix H, where for convenience we have labelled the columns of H with the corresponding directed arcs: 1 → 2 3 → 5 4 → 5 5 → 1 5 → 2  1 0 0 −1 0   −1 0 0 0 −1    H = 0 1 0 0 0  .    0 0 1 0 0  0 −1 −1 1 1 Evidently,  2 −1 0 0 −1   −1 2 0 0 −1  >   HH =  0 0 1 0 −1  ,    0 0 0 1 −1  −1 −1 −1 −1 4 which is the Laplacian matrix for the graph in Fig. 4.1. 4.2. The Laplacian Matrix 135

Suppose that we have a weighted graph G on n vertices with Laplacian matrix L. Denote the eigenvalues of L by 0 = λ1 ≤ λ2 ≤ · · · ≤ λn. From Proposition 4.2.2 (vi) and (vii), we see that λ2 > 0 if and only if G is connected. Suppose that n > > x ∈ R with x x = 1 and x 1 = 0. Let v2, . . . , vn be an orthonormal collection of eigenvectors of L corresponding to λ2, . . . , λn, respectively; note that each vj, j = 2, . . . , n, is orthogonal to 1. Then, there are scalars cj, j = 2, . . . , n, such that Pn > Pn 2 x = j=2 cjvj. Observe that 1 = x x = j=2 cj . Hence,

n n n n > X 2 > X 2 > X 2 X 2 x Lx = cj vj Lvj = cj λjvj vj = cj λj ≥ λ2 cj = λ2. j=2 j=2 j=2 j=2

> Further, we have v2 Lv2 = λ2 and, consequently, we deduce that

> > > λ2 = min x Lx|x x = 1, x 1 = 0 . (4.1)

As our next result shows, λ2 is a nondecreasing function of the weight of any edge in the graph. Proposition 4.2.4. Let G be a weighted graph and form H from G by either adding a weighted edge e to G, or increasing the weight of an existing edge in G. Denote the second smallest eigenvalues of the Laplacian matrices of G and H by λ2 and λc2, respectively. Then, λc2 ≥ λ2. Proof. Let L and Lb denote the Laplacian matrices for G and H, respectively. We > have Lb = L + w(ei − ej)(ei − ej) , where w > 0 and i ∼ j is the edge that either is added or whose weight is increased. Let y be a vector such that y>y = 1, y>1 = 0, > and y Lyb = λc2. We then have

> > 2 > > > > λc2 = y Lyb = y Ly + w(yi − yj) ≥ y Ly ≥ min y Ly|y y = 1, y 1 = 0 = λ2.

For a weighted graph G the eigenvalue λ2 of its Laplacian matrix is called the algebraic connectivity of G, and henceforth we denote it by α(G). A great deal is known about this quantity; see, for example, the surveys Kirkland [10] and Abreu [1]. Example 4.2.5. Consider the sequence of unweighted graphs shown in Fig. 4.3, which are related via edge addition. Obviously, the algebraic connectivity is nondecreasing under edge addition, though note that it is not strictly increasing, since the addition of the edge 1 ∼ 3 to the top left graph leaves the algebraic connectivity unchanged. Recall that for a connected graph that is not a complete graph, the vertex connectivity is the minimum number of vertices whose deletion (along with all incident edges) yields a disconnected graph. 136 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

1u 2 u 1u 2 u JJ JJ J J J J uJ u u u J u u 3 5 4 3 5 4 α = 1 α = 1 1u 2 u JJ J J J J J u J uJJ u 3 5 4 √ α = 3 − 2

Figure 4.3: Graphs related by edge addition and their algebraic connectivities

Proposition 4.2.6. Let G be an unweighted non-complete graph on n vertices with vertex connectivity p. Then, the algebraic connectivity of G is at most p. Proof. Suppose that, by deleting vertices n − p + 1, . . . , n from G, we get a disconnected graph G1 ∪ G2; let L1,L2 denote the Laplacian matrices for G1,G2, respectively. Adding edges to G if necessary, we ﬁnd that the algebraic connectivity of G is bounded above by that of the graph whose Laplacian matrix is   L1 + pI 0 −J L˜ ≡  0 L2 + pI −J  −J −J nI − J

(here, J denotes an all-ones matrix whose order can be determined from the context). Suppose that m1, m2 are the numbers of vertices in G1,G2 respectively. Consider the vector  m 1  1 2 x = p  −m11  . m1m2(m1 + m2) 0

Observe that x>x = 1, x>1 = 0, and x>Lx˜ = p. From (4.1), we see that the algebraic connectivity for G is bounded above by p. The following result, attributed to Kirchhoﬀ, illustrates how the algebraic properties of the Laplacian matrix can reﬂect the combinatorial properties of the underlying graph. 4.2. The Laplacian Matrix 137

Theorem 4.2.7. Suppose that G is an unweighted graph on n vertices with Laplacian matrix L. Select a pair of indices i, j between 1 and n, and let L denote the matrix formed from L by deleting the i-th row and j-th column. Then,

det(L) = (−1)i+j × the number of spanning trees in G .

Note that in Theorem 4.2.7, it does not matter which indices i and j we choose! This is the Matrix Tree Theorem, and a proof can be found in Brualdi– Ryser [5, Sect. 2.5], though we outline the main ideas of the proof here. We begin with the case when G is not connected, in which case the number of spanning trees for G is zero. For concreteness, denote the connected components of G by G1,...,Gk, with corresponding Laplacian matrices L1,...,Lk, respectively. Without loss of generality we may write L as a direct sum   L1  L2  L =   .  ..   .  Lk

Suppose that in row i of L, the nonzero entries fall in the a-th diagonal block, and in column j of L, the nonzero entries fall in the b-th diagonal block. If a = b, then L is still a direct sum of matrices, and k − 1 of those summands are singular, since they correspond to the diagonal blocks of L that were unaffected by the row and column deletion. If a =6 b, then again L has a direct sum-like structure, except that the a-th and b-th summands are rectangular, the former having more columns than rows, and the latter having more rows than columns. Evidently, L has linearly dependent columns (since one of its summands does) and so is singular. In either case, det(L) = 0, which coincides with the number of spanning trees in G. Henceforth, we take G to be connected. Recall that the adjugate of L, adj(L), is the transpose of the matrix of cofactors for L. We have adj(L)L = L adj(L) = det(L)I = 0 (see Horn–Johnson [9, Sect. 0.8.2]), so each row and column of adj(L) is a null vector for L, i.e., a multiple of the all-ones vector. It follows now that adj(L) = γJ, for some γ ∈ R. Since all of the cofactors of L are equal, it suffices to find the determinant of the leading (n − 1) × (n − 1) principal submatrix of L. Let H be an oriented incidence matrix for G, so that L = HH>. Delete the last row of H to form Hˆ ; we want to find γ = det(Hˆ Hˆ >). Using the Cauchy– Binet formula for the determinant of a product of matrices (see Horn–Johnson [9, ˆ ˆ > P ˆ ˆ > Sect. 0.8.7]), we have det(HH ) = S det(HS) det(HS ), where the sum is taken over all subsets S of n − 1 edges, and HˆS is the submatrix of Hˆ arising from the columns corresponding to S. Consider HS, the n × (n − 1) submatrix of H on the columns corresponding to S. A straightforward proof by induction on the number of vertices shows that if HS is the incidence matrix of a spanning tree then, for any (n − 1) × (n − 1) submatrix of HS, the corresponding determinant is either 1 or −1. On the other 138 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph hand, if the edges in S do not induce a spanning tree of G, then necessarily the spanning subgraph of G with edges in S is disconnected, say with t ≥ 2 connected components. For each connected component C in that subgraph, observe that the sum of the rows of HS corresponding to the vertices of C is the zero vector. It now follows that the rank of HS is at most n − t < n − 1. Hence, the rank of any (n − 1) × (n − 1) submatrix of HS is at most n − t; in particular, such a square submatrix has determinant zero. Assembling these observations, we thus find that ( 1 if S corresponds to a spanning tree, det(HˆS) = 0 if not.

ˆ ˆ > ˆ ˆ > It now follows that det(HH ) counts spanning trees, since det(HS) det(HS ) is equal to 1 if S induces a spanning tree and equal to 0 if not.

4.3 The Group Inverse

In this section we define the group inverse of a square matrix and present some basic results on the group inverse. We then consider the special case of the group inverse of a Laplacian matrix. Suppose that M is a real square matrix of order n. Suppose further that M is singular, with 0 as a semi-simple eigenvalue (i.e., the algebraic and geometric multiplicities of 0 coincide). Of course M is not invertible, but it has a group inverse, which we now define. The group inverse of M is the unique matrix X satisfying the following three properties: (i) MX = XM; (ii) MXM = M; and (iii) XMX = X. We denote this group inverse X by M #. One way of computing M # is to work with a full rank factorisation of M: if M has rank k, then there are an n × k matrix U and a k × n matrix V such that M = UV ; see Horn–Johnson [9, Sect. 0.4]. Since M has k nonzero eigenvalues then, recalling that the matrices M = UV and VU have the same nonzero eigenvalues, we find that the k × k matrix VU has k nonzero eigenvalues. Hence, VU is invertible. In that case, the matrix X = U(VU)−2V is easily seen to satisfy properties (i)–(iii). Next, we address the uniqueness issue. Suppose that we have matrices X1,X2 both satisfying properties (i)–(iii). We find that MX1MX1 = MX1, so MX1 is a projection matrix. Letting col(·) and N(·) denote the column space and null space of a matrix, respectively, we have col(MX1) ⊆ col(M) and rank(M) = rank(MX1M) ≤ rank(MX1) ≤ rank(M), so rank(MX1) = rank(M). We thus deduce that col(MX1) = col(M). Similarly we deduce that N(MX1) = N(M), so that MX1 is the projection matrix with range col(M) and null space N(M). Similarly, MX2 is the projection matrix with range col(M) and null space N(M). Recall that, given any n × n projection matrix P , and any vector x ∈ Rn, we may write x as x = y + z, where y ∈ col(P ), z ∈ N(P ); see Halmos [8, Sect. 41]. It now follows that two projection matrices with the same column space and the 4.3. The Group Inverse 139

same null space must be equal. We thus deduce that MX2 = MX1. Note also that, by (i), we also have X2M = X1M. But then we have

X1 = X1MX1 = X1(MX2) = (X1M)X2 = (X2M)X2 = X2, so that there is a unique matrix satisfying (i)–(iii). Hence, we have that M # = U(VU)−2V . Remark 4.3.1. Consider the special case in which 0 is an algebraically simple eigenvalue of M, say with u and v> as right and left null vectors, normalized so that v>u = 1. We claim that in this case, X is the group inverse of M if and only if MX = XM = I − uv>, Xu = 0, and v>X = 0>. To see the claim, suppose ﬁrst that X satisﬁes (i)–(iii). Since M(XM −I) = 0, each column of XM −I is a scalar multiple of u. Also, (MX − I)M = 0, so each row of MX − I is a scalar multiple of v>. Hence, XM = I + uw> for some vector w and MX = I + zv> for some vector z. But XM = MX, so it must be the case that XM = MX = I + tuv> for some scalar t. Since det(XM) = 0, it follows that t = −1. For the converse, suppose that MX = XM = I − uv>, Xu = 0, and v>X = 0>. Then, MXM = (I − uv>)M = M − uv>M = M, while XMX = X(I − uv>) = X − xuv> = X, so that properties (ii)–(iii) hold for the matrix X, while property (i) holds by hypothesis. Hence X = M #. Example 4.3.2. Consider the n × n matrix " # In−1 −1n−1 M = > . −1n−1 n − 1 Evidently, M is the Laplacian matrix of the unweighted star on n vertices, so it has nullity 1 with left and right null vectors 1> and 1, respectively. Let

" 2 # 1 n In−1 − (n + 1)J −1n−1 X = 2 > n −1n−1 n − 1 and observe that X1 = 0 and 1>X = 0>. It is straightforward to determine that

" 2 # 1 n In−1 − nJ −n1n−1 MX = 2 > = XM, n −n1n−1 n(n − 1)

1 > so that MX = XM = I − n 11 . Appealing to Remark 4.3.1, we ﬁnd that M # = X. Next we give some alternative expressions for the group inverse of the Lapla- cian matrix of a graph. Theorem 4.3.3. Let L be the Laplacian matrix of a connected weighted graph G on n vertices. Denote the eigenvalues of L by 0 = λ1 < λ2 ≤ · · · ≤ λn, and let Q be an > orthogonal matrix (of eigenvectors) such that L = Q diag 0 λ2 . . . λn Q . Then, # 0 1 ... 1 >. L = Q diag λ2 λn Q 140 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

3u4 u 5 u HH ¨¨ JJ H JJ ¨ J HH J¨¨ H ¨ J ¨¨HHJ Ju¨ HJ u 1 2

Figure 4.4: Graph for Example 4.3.4

Proof. 0 1 ... 1 > Setting X = Q diag λ2 λn Q , it is readily veriﬁed that XL = LX, LXL = L, and XLX = X. Example 4.3.4. Consider the graph depicted in Fig. 4.4, and observe that its Lapla- cian matrix is given by

 4 −1 −1 −1 −1   −1 4 −1 −1 −1    L =  −1 −1 3 −1 0  .    −1 −1 −1 3 0  −1 −1 0 0 2

It can be veriﬁed that the following orthogonal matrix Q diagonalizes L:

0 2 4 5 5  √1 √1 √3  5 0 0 2 30  √1 0 0 √−1 √3   5 2 30     √1 √1 √1 0 √−2  Q = 5 6 2 30     √1 √1 √−1 0 √−2   5 6 2 30  √1 √−2 √−2 5 6 0 0 30

(here, above each column of Q we have listed the eigenvalue of L for which that column is an eigenvector). Using Theorem 4.3.3 we can, for instance, compute the (3, 5) entry of L# as

> 1 1 1 1 > e3 Q diag 0 2 4 5 5 Q e5  √1  5  √−2  h i  6  −7 √1 √1 √1 0 √−2 1 1 1 1   = 5 6 2 30 diag 0 2 4 5 5  0  = .   50  0  √−2 30 We have the following consequence of Theorem 4.3.3. 4.4. L# and the Bottleneck Matrix 141

Corollary 4.3.5. Suppose that L is as in Theorem 4.3.3 and that t =6 0 is given. # t −1 1 Then, L = (L + n J) − tn J. t > t −1 Proof. Observe that n J = Q diag t 0 ··· 0 Q , so that (L + n J) − 1 −1 > 1 > tn J = Q diag t λ2 ··· λn Q −Q diag t 0 ··· 0 Q . The conclusion now follows readily. Example 4.3.6. Here, we revisit the Laplacian matrix L for the unweighted graph in Fig. 4.1. We have  2 −1 0 0 −1   3 0 1 1 0   −1 2 0 0 −1   0 3 1 1 0       0 0 1 0 −1  and L + J =  1 1 2 1 0  ,      0 0 0 1 −1   1 1 1 2 0  −1 −1 −1 −1 4 0 0 0 0 5 so that  7 2 1 1  15 15 − 5 − 5 0  2 7 1 1   15 15 − 5 − 5 0    −1  − 1 − 1 4 − 1  (L + J) =  5 5 5 5 0  .    − 1 − 1 − 1 4 0   5 5 5 5  1 0 0 0 0 5 We now ﬁnd from Corollary 4.3.5 that  32 7 6 6 1  75 75 − 25 − 25 − 25  7 32 6 6 1   75 75 − 25 − 25 − 25  1   # −1 −  − 6 − 6 19 − 6 − 1  L = (L + J) J =  25 25 25 25 25  . 25    − 6 − 6 − 6 19 − 1   25 25 25 25 25  1 1 1 1 4 − 25 − 25 − 25 − 25 25

4.4 L# and the Bottleneck Matrix

Evidently L, and hence L#, carries information about the associated weighted graph G. Our goals in this chapter are to (i) determine how the combinatorial structure of G is reﬂected in L#, and (ii) use L# to get information about G. The results in this section will assist in making progress toward both goals. Theorem 4.4.1. Suppose that G is a connected weighted graph on n vertices with Laplacian matrix L. Denote the leading principal submatrix of L of order n − 1 by −1 L{n}, and let B = L{n}. Then,

> " 1 1 1 # # 1 B1 B − n BJ − n JB − n B1 L = 2 J + 1 > . n − n 1 B 0 142 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

Proof. Note that L can be written as " # L{n} −L{n}1 L = > > . −1 L{n} 1 L{n}1

This yields the full rank factorization L = UV , where " # L{n} −1 U = > ,V = I . −1 L{n}

−1 −1 −1 1 Note that VU = (I + J)L{n}, so that (VU) = L{n}(I + J) = B(I − n J). The expression for L# now follows from the fact that L# = U(VU)−2V . Example 4.4.2. Again we consider the unweighted graph in Fig. 4.1, whose Lapla- cian matrix is given by

 2 −1 0 0 −1   −1 2 0 0 −1    L =  0 0 1 0 −1  .    0 0 0 1 −1  −1 −1 −1 −1 4

In the notation of Theorem 4.4.1, we have

 2 1  3 3 0 0  1 2   3 3 0 0  B =   ,  0 0 1 0    0 0 0 1 and it now follows that  32 7 6 6 1  75 75 − 25 − 25 − 25  7 32 − 6 − 6 − 1   75 75 25 25 25  #  6 6 19 6 1  L =  − − − −  .  25 25 25 25 25   6 6 6 19 1   − 25 − 25 − 25 25 − 25  1 1 1 1 4 − 25 − 25 − 25 − 25 25 Of course this agrees with the conclusion of Example 4.3.6. Matrix B in Theorem 4.4.1 is known as the bottleneck matrix based at vertex n for the weighted graph. Using the adjugate formula for the inverse (see Horn– Johnson [9, Sect. 0.8]), we see that, for each i, j = 1, . . . , n − 1,

i+j det(L{j,n},{i,n}) bi,j = (−1) , (4.2) det(L{n}) 4.4. L# and the Bottleneck Matrix 143

where L{j,n},{i,n} is the (n − 2) × (n − 2) matrix formed from L by deleting rows j, n and columns i, n. Earlier, we saw in Theorem 4.2.7 that for an unweighted graph, det(L{n}) counts the number of spanning trees in the graph – in fact a more general result, the All Minors Matrix Tree Theorem (see Chaiken [6]), will assist us in interpreting the entries of the bottleneck matrix B. Suppose that we have an n × n matrix M such that (a) mi,j ≤ 0 whenever i =6 j and (b) the column sums of M are all zero. (Note that M does not have to be symmetric here.) For each i =6 j we think of |mi,j| as the weight of the arc i → j in the loopless directed graph associated with M. Let W, U be subsets of {1, . . . , n} of the same cardinality, say k. Consider the (n − k) × (n − k) matrix Mc formed from M by deleting the rows indexed by W and the columns indexed by U. Then det(Mc) can be computed in the following manner. Consider the spanning directed forests F in the loopless directed graph of M such that (i) F has exactly k trees; (ii) each tree in F contains precisely one vertex in U and one vertex in W ; and (iii) every arc F is directed away from the vertex in U (of the tree containing that arc). For each such F , let w(F ) denote the product of the weights of the arcs in F . Then, det(Mc) is equal to a sum of P the type F w(F ), where the sum is taken over all directed forests satisfying (i)–(iii), and where the coefficient is determined by W , U, and F . Here are the details of how this coefficient is determined. Note that since each tree of F contains precisely one vertex in W and one vertex in U, there is a bijection f : W → U with f(i) = j precisely in the case that i and j are in the same tree of F ; set p(f) = |{{a, b} ⊂ W | a < b, f(a) > f(b)}| (this is the number of inversions in f). We also define q(W ) = |{{a, b} | a < b, a ∈ {1, . . . , n}\ W, b ∈ W }|; q(U) is defined analogously. The coefficient is then given by (−1)q(W )+q(U)+p(f), so that X det(Mc) = (−1)q(W )+q(U) (−1)p(f)w(F ). (4.3) F

In the special case that W = U (i.e., we are looking at a principal minor of M), P we have p(f) = 0 and q(W ) = q(U), so that det(Mc) = F w(F ). Next, we focus on the case in which G is an undirected connected weighted graph on n vertices, and L is its Laplacian matrix. Denote the set of spanning trees {i,j} of G by S, and for each i, j, k = 1, . . . , n, we let Sk denote the set of spanning forests of G consisting of just two trees, one of which contains vertex k and the other of which contains vertices i and j. With this in hand, we have the following facts, both of which are consequences of the All Minors Matrix Tree Theorem (as usual, we interpret an empty sum as 0): P Fact 1: det(L{n}) = T ∈S w(T ). This is because, in the notation above, we are taking W = U = {n}, and the spanning directed forests of interest are in one-to- one correspondence with the spanning trees in S. i+j P Fact 2: det(L{j,n},{i,n}) = (−1) {i,j} w(F). Here, we have W = {j, n} and F∈Sn 144 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

U = {i, n}, so that the spanning directed forests of interest are in one-to-one {i,j} {i,j} correspondence with the forests in Sn . For each forest in Sn , the corresponding bijection f is given by f(j) = i, f(n) = n, so that p(f) = 0. Further, q(W ) = j + n − 3 and q(U) = i + n − 3, so (4.3) now yields det(L{j,n},{i,n}) = i+j P (−1) {i,j} w(F). F∈Sn We refer the interested reader to Chaiken [6] for a full discussion of the All Minors Matrix Tree Theorem. Observe that the amount of work required in order to compute the entries of the bottleneck matrix via Facts 1 and 2 above is a function of the number {i,j} of spanning trees in the graph, and the cardinality of the set Sn . Thus this approach to the bottleneck matrix is most tractable when the graph in question is sparse. We remark that for each k = 1, . . . , n, we can analogously deﬁne the bottleneck matrix based at vertex k as the inverse of L{k}, the principal submatrix of the Laplacian formed by deleting its k-th row and column. Thus, an analogue of Theorem 4.4.1 using the bottleneck matrix based at any vertex k also holds. Example 4.4.3. We revisit Example 4.4.2. Since the graph in question (depicted in Fig. 4.1) is unweighted, every edge weight is equal to 1. Our graph has three spanning trees (each formed by deleting one edge of the three-cycle through ver- {1,1} tices 1, 3, and 5), so det(L{5}) = 3. Observe that S5 consists of two forests: the one formed by deleting the edges 1 ∼ 5 and 1 ∼ 2, and the one formed by {1,2} deleting the edges 1 ∼ 5 and 2 ∼ 5. Hence, det(L{1,5},{1,5}) = 2. Similarly, S5 contains a single forest, formed by deleting the edges 1 ∼ 5 and 2 ∼ 5, so that {1,j} det(L{1,5},{2,5}) = −1. Noting that for j = 3, 4, S5 = ∅, it now follows that the 2 1 ﬁrst row of the bottleneck matrix based at vertex n is given by 3 3 0 0 . Analogous arguments can be used to determine the rest of the entries in that bottleneck matrix.

4.5 L# for Weighted Trees

In this section we consider the group inverse of the Laplacian matrix for a weighted tree. As we will see, the simple combinatorial structure of trees makes the interpretation of the entries in the group inverse especially transparent. We begin by establishing a simple expression for the entries in the bottleneck matrix. Theorem 4.5.1. Let T be a weighted tree on n vertices, and let B denote the corresponding bottleneck matrix based at vertex n. For each index k between 1 and n − 1, let Pk,n denote the path from vertex k to vertex n in T . Then, for each − , P . i, j = 1, . . . , n 1 bi,j = e∈Pi,n∩Pj,n 1/w(e) Proof. Fix indices i, j between 1 and n − 1, and note that the spanning forests with exactly two trees are precisely those graphs that arise by deleting a single {i,j} edge from T . It now follows that a spanning forest F is in Sn if and only if it is 4.5. L# for Weighted Trees 145

1u 2 u e1 e2

u u u u u 3e3 4e4 5e5 6e6 7

Figure 4.5: Tree for Example 4.5.2

formed from T by deleting a single edge, say eF ∈ Pi,n ∩ Pj,n, from T . Observing that the weight of such a forest is w(F ) = w(T )/w(eF ), the conclusion now follows from (4.2) and Facts 1 and 2.

Observe that in Theorem 4.5.1, the set Pi,n ∩ Pj,n consists of the edges that are simultaneously on the path from i to n and on the path from j to n. Informally, we might think of |Pi,n ∩ Pj,n| as the number of “bottlenecks” on the route from i to n and on the route from j to n, hence the term bottleneck matrix. We can represent the bottleneck vertex based at vertex n for a weighted tree T in an alternate format, as follows. Denote the edge set of T by E, and for each edge e ∈ E, deﬁne the vector u(e) ∈ Rn−1 via

( 1, if j is in the connected component of T \ e without vertex n u(e)j = 0, otherwise.

It then readily follows from Theorem 4.5.1 that the bottleneck matrix based at P 1 > vertex n is given by B = e∈E w(e) u(e)u(e) .

Example 4.5.2. Consider the unweighted tree in Fig. 4.5 where for convenience we have labelled the edges e1, . . . , e6. It is straightforward to determine that the bottleneck matrix based at vertex 7 is given by

 3 1 2 2 2 1   1 2 1 1 1 1     2 1 4 3 2 1  B =   .  2 1 3 3 2 1     2 1 2 2 2 1  1 1 1 1 1 1 146 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

P6 > Below we illustrate how B can be expressed in the form j=1 u(ej)u(ej) :

e1 e2 e3  1 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0   0 1 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   0 0 1 0 0 0    +   +    0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

e4 e5 e6  0 0 0 0 0 0   1 0 1 1 1 0   1 1 1 1 1 1   0 0 0 0 0 0   0 0 0 0 0 0   1 1 1 1 1 1         0 0 1 1 0 0   1 0 1 1 1 0   1 1 1 1 1 1  +   +   +  .  0 0 1 1 0 0   1 0 1 1 1 0   1 1 1 1 1 1         0 0 0 0 0 0   1 0 1 1 1 0   1 1 1 1 1 1  0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 Here, each rank-one summand is labeled with the corresponding edge. Next, we want to develop some alternative expressions for the entries in L# when L is the Laplacian matrix of a weighted tree on n vertices. We introduce the following notation: for each edge e in the edge set E of T , and any vertex k = 1, . . . , n, let βk(e) denote the set of vertices in the connected component of T \ e not containing vertex k. Observe that with this notation and any edge e, > 1 u(e) = |βn(e)|. From Theorem 4.4.1, we ﬁnd that > 2 1 B1 1 X 1 1 X |βn(e)| l# = = (1>u(e))2 = . n,n n2 n2 w(e) n2 w(e) e∈E e∈E Next we consider oﬀ-diagonal entries in L#. Fix an index i =6 n, and consider # # # 1 > li,n. From Theorem 4.4.1, li,n = ln,n − n ei B1. Note that

> X 1 e B1 = u(e)i|βn(e)|. i w(e) e∈E

Observing that u(e)i = 1 if and only if e ∈ Pi,n, and 0 otherwise, we ﬁnd that > 1 P 1 | | ei B = e∈Pi,n w(e) βn(e) . Assembling the above, it follows that 2 1 X |βn(e)| 1 X |βn(e)| l# = − . i,n n2 w(e) n w(e) e∈E e∈Pi,n As noted above, we can construct L# using the bottleneck matrix based at any vertex of our graph. Consequently, we have the following. For each k = 1, . . . , n, 2 1 X |βk(e)| l# = k,k n2 w(e) e∈E 4.5. L# for Weighted Trees 147 and, for any i, j = 1, . . . , n − 1 with i =6 j,

2 1 X |βj(e)| 1 X |βn(e)| l# = − . i,j n2 w(e) n w(e) e∈E e∈Pi,j

We have one more alternate expression for L# for a weighted tree T . For each edge e of T , denote the connected components of T \ e (the graph formed n from T by deleting edge e) by T1(e) and T2(e). Let v(e) be the vector in R given by ( |βj (e)| n , if j ∈ T1(e), v(e)j = (4.4) |βj (e)| − n , if j ∈ T2(e), for j = 1, . . . , n. Theorem 4.5.3. If L is the Laplacian matrix of a weighted tree with edge set E, then X 1 L# = v(e)v(e)>. w(e) e∈E > Proof. For each k = 1, . . . , n note that, since ek v(e) = |βk(e)|/n, 2 1 X |βk(e)| X 1 l# = = (e>v(e))2. k,k n2 w(e) w(e) k e∈E e∈E Next suppose that we have indices i, j with i =6 j. Fix an edge e of T , and suppose that e∈ / Pi,j. Then we have i, j ∈ T1(e) or i, j ∈ T2(e) and in either case, > > 2 2 2 ei v(e)v(e) ej = |βi(e)||βj(e)|/n = |βj(e)| /n , since βi(e) = βj(e) in this case. Next, we suppose that e ∈ Pi,j, so that i ∈ T1(e), j ∈ T2(e), or vice versa. In this > > 2 2 case, we have ei v(e)v(e) ej = −|βi(e)||βj(e)|/n = −(n − |βj(e)|)|βj(e)|/n = 2 2 |βj(e)| /n − |βj(e)|/n. By summing over the edges in T , it now follows that # P 1 > > li,j = e∈E w(e) ei v(e)v(e) ej. For a weighted tree on n vertices, we ﬁnd from Proposition 4.2.2 (ii) that its Laplacian matrix can be written as a sum of n − 1 matrices, each of which corresponds to a particular edge; each summand consists of a scalar multiple (determined by the weight of the edge) of a symmetric rank-one matrix corresponding to the associated edge. We note that Theorem 4.5.3 has a similar ﬂavor, as it ex- presses L# as a sum of n − 1 matrices, one for each edge in the tree; further, each summand is an edge-weight-determined scalar multiple of a symmetric rank-one edge-determined matrix. Theorem 4.5.3 yields the following. Corollary 4.5.4. Let T be a weighted tree with edge set E and vertices 1, . . . , n.

# 1 P |βk(e)|(n−|βk(e)|) (i) Fix any vertex k between 1 and n. Then, trace(L )= n e∈T w(e) . For any pair of vertices , let P . Then, # (ii) i, j δ(i, j) = e∈Pi,j 1/w(e) trace (L ) = 1 P n 1≤i

Proof. (i). Observe that

X 1 X 1 trace (L#) = trace (v(e)v(e)>) = v(e)>v(e). w(e) w(e) e∈E e∈E

> Noting that, for any vertex k of T , v(e) v(e) = |βk(e)|(n − |βk(e)|)/n, the conclusion follows. P P P ∈ (ii). Observe that 1≤i

X X X 1 X |βk(e)|(n − |βk(e)|) δ(i, j) = = w(e) w(e) 1≤i

Example 4.5.5. Consider the unweighted path Pn on n vertices, with vertex 1 adjacent to vertex 2, vertex j adjacent to vertices j−1 and j+1, for j = 2, . . . , n−1, and vertex n adjacent to vertex n − 1. For each k = 1, . . . , n − 1, let e(k) denote the edge between vertices k and k + 1. Then,

" (n−k)2 k(n−k) # > n2 Jk − n2 Jk,n−k v(e(k))v(e(k)) = k(n−k) k2 . − n2 Jn−k,k n2 Jn−k

> > 2 2 It now follows that, for each j = 1, . . . , n−1, ej v(e(k))v(e(k)) ej = (n−k) /n if > > 2 2 j ≤ k, while ej v(e(k))v(e(k)) ej = k /n if j ≥ k + 1. Next, suppose that i < j. Then,  (n−k)2 2 if i < j ≤ k,  n > >  k(n−k) ei v(e(k))v(e(k)) ej = − n2 if i ≤ k < j,   k2 n2 if k + 1 ≤ i < j. Obviously, analogous relations hold when i > j. A couple of computations now show that if L is the Laplacian matrix for Pn, then

 j(j−1)(2j−1)+(n−j)(n−j+1)(2n−2j+1) |j−i|(j+i−1)  6n2 − 2n if i ≤ j l# = i,j i(i−1)(2i−1)+(n−i)(n−i+1)(2n−2i+1) |j−i|(j+i−1)  6n2 − 2n if i ≥ j + 1.

4.6 Algebraic Connectivity

Throughout this section, we continue to denote the algebraic connectivity of a weighted graph G by α(G). Our goal is to provide bounds on α(G) in terms of the group inverse of the corresponding Laplacian matrix. We remark here that, while 4.6. Algebraic Connectivity 149 the algebraic connectivity does reflect the structure of the graph, it can do so in a rather complicated fashion. Producing bounds on the algebraic connectivity can provide insight into how graph structure influences the algebraic connectivity. For any n × n real matrix M whose row sums are constant, we define the > function τ(M) by τ(M) = (maxi,j=1,...,n k(ei − ej) Mk1)/2. The following result shows how we can use τ to bound Laplacian eigenvalues. Lemma 4.6.1. Let G be a connected weighted graph on n vertices with Laplacian matrix L. Then, 1 (i) α(G) ≥ τ(L#) ; n−1 (ii) α(G) ≤ # . n maxj=1,...,n lj,j Proof. We will begin with the proof of a general observation, then apply the observation to prove (i) and (ii). Suppose that M is a symmetric matrix of order n with constant row sums, say with M1 = r1. Let λ =6 r be an eigenvalue of M; we intend to show that |λ| ≤ τ(M). To that end, let z be a real eigenvector of M > such that ||z||1 = 1. Observe that z 1 = 0. We claim that there are scalars a(i, j), 1 ≤ i < j ≤ n such that X X ei − ej |a(i, j)| = ||z||1 and z = a(i, j) . 2 1≤i

To see the claim note that, without loss of generality, we may write

> z = z1 ··· zk −zk+1 · · · −zn , where z1 ≥ zj ≥ 0, for j = 1, . . . , n, and k is some suitably chosen index. We now > 1 > > write z = 2zk+1 2 (e1 − ek+1) +z ˆ , where

> zˆ = (z1 − zk+1) z2 ··· zk 0 −zk+2 · · · −zn .

Note that ||zˆ||1 = ||z||1 − 2zk+1, and thatz ˆ has fewer nonzero entries than z. We can now proceed by induction on the number of nonzeros in z in order to establish the claim; see Seneta [14, Sect. 2.5] for further details. Applying the claim, we have

> > X ei − ej |λ| = |λ|||z||1 = ||z M||1 = a(i, j) M 2 1 1≤i

> X ei − ej X ≤ |a(i, j)| M ≤ |a(i, j)|τ(M) = ||z||1τ(M), 2 1 1≤i

(i) Consider L#, which has constant row sums (equal to 0) and 1/α(G) as a nonzero eigenvalue. From the claim, 1/α(G) ≤ τ(L#), and the desired inequality follows. (ii) Denote the nonzero eigenvalues of L by α(G) ≡ λ2 ≤ λ3 ≤ · · · ≤ λn, and let v2, . . . , vn denote an orthonormal collection of corresponding eigenvectors. # Pn 1 > Pn > 1 Then, by Theorem 4.3.3, L = vkv . Note that vkv = I − J. k=2 λk k k=2 k n Fix an index j between 1 and n. Then, n n # X 1 > > 1 X > 2 n − 1 lj,j = ej vkvk ej ≤ (ej vk) = . λk α(G) nα(G) k=2 k=2 The conclusion now follows. Next, we apply Lemma 4.6.1 to weighted trees. For any weighted tree, we let P(T ) denote the set of pendent vertices of T , i.e., the set of vertices of degree 1. Theorem 4.6.2. Let T be a weighted tree on n vertices. Then, n n o ≤ α(T ) P |βi(e)|(n−|βi(e)|) | ∈ P max e∈Pi,j w(e) i, j (T ) n(n − 1) ≤ min 2 | i ∈ P(T ) . P |βi(e)| e∈T w(e) Proof. First we consider the lower bound on α(T ). In view of Lemma 4.6.1 (i), it suﬃces to show that   1  X |βi(e)|(n − |βi(e)|)  τ(L#) ≤ max | i, j ∈ P(T ) . n w(e) e∈Pi,j 

# P 1 > Applying Theorem 4.5.3, we have L = e∈T w(e) v(e)v(e) , where v(e) is as deﬁned in (4.4). Observe that, for any edge e and indices i, j, ( > 1 if e ∈ Pi,j, (ei − ej) v(e) = 0 if e∈ / Pi,j.

Since ||v(e)||1 = 2|βi(e)|(n − |βi(e)|)/n whenever e ∈ Pi,j, we ﬁnd that

> # X v(e) ||(ei − ej) L ||1 = w(e) 1 e∈Pi,j

X ||v(e)||1 ≤ w(e) e∈Pi,j X 2 = |βi(e)|(n − |βi(e)|). nw(e) e∈Pi,j 4.6. Algebraic Connectivity 151

It is straightforward to determine that the rightmost expression above is maxi- mized for some pair of indices i, j ∈ P(T ). The lower bound on α(T ) now follows. Next we consider the upper bound on α(T ). From Lemma 4.6.1 (ii) we have n−1 α(G) ≤ # , and note that, for any index j, n maxj=1,...,n lj,j

> 2 2 X (ej v(e)) X |βj(e)| l# = = . j,j w(e) n2w(e) e∈T e∈T

Consequently, for any j we have n(n − 1) α(T ) ≤ 2 . P |βj (e)| e∈T w(e)

# Next, we claim that lj,j is maximised at some pendent vertex j. (Once the claim is established, the upper bound on α(T ) follows readily.) To see the claim, we suppose that vertex k is not pendent, say with k adjacent to vertices m1 and m2. Let e, eˆ denote the edges between vertex k and vertices m1, m2, respectively. # ≥ # | | ≥ | | − | | Observe that lk,k lm1,m1 if and only if βk(e) βm1 (e) = n βk(e) , i.e., if | | ≥ # ≥ # | | ≥ and only if βk(e) n/2. Similarly, lk,k lm2,m2 if and only if βk(ê) n/2. Since βk(e) ∩ βk(ê) = ∅ and neither set contains the vertex k, it must be the case # that |βk(e)|+|βk(ê)| ≤ n−1. We conclude then that lk,k must be strictly less than # # # one of lm1,m1 and lm2,m2 . It now follows that lj,j is maximised by some pendent vertex j, as claimed.

Example 4.6.3. Consider the (unweighted) star on n ≥ 3 vertices, K1,n−1. Fix a pendent vertex i, and note that ( n − 1 if e is incident with i, |βi(e)| = 1 if e is not incident with i.

P | |2 − 2 − 2 − − We deduce that e∈K1,n−1 βi(e) = (n 1) + n 2 = n n 1. Thus the 2 2 upper bound in Theorem 4.6.2 is α(K1,n−1) ≤ (n − n)/(n − n − 1). Recall from Example 4.3.2 that the Laplacian matrix for the star is given by " # In−1 −1n−1 L = > . −1n−1 n − 1

It is straightforward to see that the eigenvalues of L are: 0 (with corresponding eigenvector 1), 1 (of multiplicity n − 1, and with (e1 − ej), j = 2, . . . , n as a −1 collection of linearly independent eigenvectors), and n (with n−1 as an n − 1 eigenvector). In particular, α(K1,n−1) = 1, so we see that in this case the upper bound in Theorem 4.6.2 is reasonably accurate for large values of n. 152 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

3u4 u 5 u HH ¨¨ JJ H JJ ¨ J HH J¨¨ H ¨ J ¨¨HHJ Ju¨ HJ u 1 2

Figure 4.6: (K2 ∪ K1) ∨ K2

Example 4.6.4. Take Pn to be the unweighted path on n vertices, say with 1 and n as the pendent vertices, and vertex j adjacent to vertex j + 1, j = 1, . . . , n − 1. P | | − | | Pn−1 − 2 − We have e∈P1,n β1(e) (n β1(e) ) = k=1 k(n k) = n(n 1)/6. Hence, the 2 lower bound on α(Pn) in Theorem 4.6.2 is 6/(n − 1) ≤ α(Pn). jπ The eigenvalues of the Laplacian matrix for Pn are given by 2(1 − cos( n )), j = 0, . . . , n−1. This is perhaps most easily seen by observing that for each such j, the vector [cos(jπ/2n), cos(3jπ/2n), cos(5jπ/2n),..., cos((2n − 1)jπ/2n)]> is an eigenvector corresponding to 2(1 − cos(jπ/n)); this is veriﬁed by making use of the identity cos((k − 1)jπ/n) + cos((k + 1)jπ/n) = 2 cos(kjπ/n), for each k ∈ N. 2 2 Consequently, α(Pn) = 2(1 − cos(π/n)), which is asymptotic to π /n as n → ∞. So, for this example we see that for large n, the lower bound in Theorem 4.6.2 is of the same order of magnitude as the true value, but is about 0.6α(Pn).

4.7 Joins

Suppose that we have two unweighted graphs G1,G2 on k and l vertices, respectively. The join of G1 and G2, denoted G1 ∨ G2, is the graph on k + l vertices formed from G1 ∪ G2 by adding in all possible edges between vertices of G1 and vertices of G2. Fig. 4.6 depicts the join of K2 ∪ K1 and K2. (We note in passing that we saw this graph earlier in Example 4.3.4.) In this section we show how joins of certain graphs yield equality in the conclusion of Lemma 4.6.1 (i). Observe that if the Laplacian matrices for G1 and G2 are L1 and L2 respectively, then the Laplacian matrix of G1 ∨ G2 can be written as L + lI −J L = 1 . −J L2 + kI

In this setting, we can ﬁnd L# from Corollary 4.3.5 (with t chosen as k + l) as follows: 1 L# = (L + J)−1 − J (k + l)2 −1 L1 + lI + J 0 1 = − 2 J 0 L2 + kI + J (k + l) 4.7. Joins 153

−1 " −1 1 # (L1 + lI) − l(k+1) J 0 1 = −1 1 − 2 J. 0 (L2 + kI) − k(k+l) J (k + l) It turns out that no characterization of the equality case in Lemma 4.6.1 (i) is known. The next result supplies a family of examples for which equality holds in that bound.

Theorem 4.7.1. Suppose that G is an unweighted graph of the form H ∨ Kp, where H is not connected. Denoting the Laplacian matrix for G by L, we have α(G) = 1/τ(L#) = p.

Proof. For concreteness, suppose that H has q vertices. Let L1 denote the Lapla- cian matrix for H, and note that L can be written as

L + pI −J L = 1 . −J (p + q)I − J

Then, L + pI + J 0 L + J = 1 , 0 (p + q)I and it follows that

" −1 1 # # (L1 + pI) − p(p+q) J 0 1 L = 1 − 2 J. 0 p+q I (p + q)

In particular, observe that τ(L#) = τ(L + J)−1. −1 −1 Next, we claim that τ(L + J) = 1/p. It turns out that (L1 + pI) is entrywise nonnegative, that its row sums are all equal to 1/p, and further that −1 since H is not connected, in fact (L1 + pI) is a block diagonal matrix. It now > −1 follows that for 1 ≤ i, j ≤ q, k(ei − ej) (L + J) k1 ≤ 2/p, with equality holding for at least one such pair i, j. Further, if 1 ≤ i ≤ q < j ≤ p + q, we have > −1 k(ei − ej) (L + J) k1 ≤ 1/p + 1/(p + q) < 2/p. Finally, if q + 1 ≤ i, j ≤ p + q, > −1 −1 k(ei − ej) (L + J) k1 = 2/(p + q) < 2/p. It now follows that τ(L + J) = 1/p, as claimed. Applying Lemma 4.6.1 (i), we thus ﬁnd that α(G) ≥ 1/τ(L#) = p. Next, noting that the vertex connectivity of G is at most p and applying Proposition 4.2.6, we ﬁnd that α(G) ≤ p. The conclusion now follows.

Example 4.7.2. Set S1 = {Kp | p ∈ N}, and for each m ∈ N, deﬁne Sm+1 = {G ∨ Kp | G ∈ Sm, p ∈ N}, where G denotes the complement of G. The collection S = ∪m∈NSm is the family of connected threshold graphs, about which much is known; see Mahadev–Peled [12]. For instance, a typical graph in S3 has the form (Kr ∪Kq)∨Kp. (Observe that the graph in Fig. 4.6 is in S3.) A straightforward application of Theorem 4.7.1 shows that, for any connected threshold graph, equality holds in the inequality of Lemma 4.6.1 (i). 154 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph 4.8 Resistance Distance

In this section we will see how the group inverse of the Laplacian matrix gives rise to a distance function for weighted graphs. Let G be a connected weighted graph on n vertices with Laplacian matrix L. Fix indices i, j, and define the resistance # # # distance from i to j, r(i, j) as r(i, j) = li,i + lj,j − 2li,j. One might wonder why the term “resistance distance” is appropriate here; the first result of this section provides a partial explanation by establishing that r(i, j) can be interpreted as a distance function. Theorem 4.8.1. (i) r(i, j) ≥ 0, with equality if and only if i = j; (ii) r(i, j) = r(j, i) for all i, j; (iii) for any i, j, k, r(i, j) ≤ r(i, k) + r(k, j), with equality holding if and only if either k coincides with one of i and j, or every path from i to j passes through vertex k. Proof. (i). Consider the case i =6 j and, without loss of generality, suppose that > # j = n. Observe that r(i, n) = (ei − en) L (ei − en). From Theorem 4.4.1, we find that " # B − 1 BJ − 1 JB − 1 B1 − > n n n − r(i, n) = (ei en) 1 > (ei en), − n 1 B 0 where B is the bottleneck matrix for G based at vertex n. We thus deduce that r(i, j) = bi,i > 0. (ii). Obvious from the definition. (iii) The desired inequality is equivalent to

# # # # 0 ≤ lk,k + li,j − li,k − lk,j. (4.5)

Evidently, both sides of (4.5) are zero if k is either i or j, so henceforth we assume that k =6 i, j. Taking k = n without loss of generality, and letting B denote the bottleneck matrix for G based at vertex n, we find from Theorem 4.4.1 that 1 > 1 > 1 > 1 > (4.5) holds if and only if 0 ≤ bi,j − n ei B1 − n 1 Bej + n ei B1 + n 1 Bej. Thus (4.5) is equivalent to the inequality bi,j ≥ 0, and it now follows that r(i, j) ≤ r(i, n)+r(n, j). Inspecting the argument above, we see that r(i, j) = r(i, n)+r(n, j) if and only if bi,j = 0. From Fact 2 in section 4.4, the latter holds if and only if {i,j} Sn = ∅, i.e., if and only if every path from i to j goes through vertex n. Having shown that the resistance distance is indeed a distance function, we now make the connection with the notion of resistance. Consider a weighted graph, which we associate with a network of electrical resistors; each edge e has an associated resistance ρ(e) ≡ 1/w(e). Fix a pair of indices i, j and suppose that we allow current to enter at vertex i and exit at vertex j. Set the voltage at i to be 1 and the voltage at j to be 0. The effective resistance between i and j is defined as the reciprocal of the current flowing in at vertex i. Using Kirchhoff’s laws and 4.8. Resistance Distance 155

1u 2 u 1u 2 u Q Q Qu 5 u u u u 3 4 3 4 r(1, 2) = 3/4 r(1, 2) = 6/11

Figure 4.7: Resistance distance for two related graphs

Ohm’s law, it turns out that this effective resistance is given by the diagonal entry corresponding to vertex i of the bottleneck matrix based at vertex j, i.e., it is equal to our resistance distance r(i, j); the details of that derivation are given in Kirkland–Neumann [11, Sect. 7.5]. Thus, the use of the word “resistance” in describing r(i, j) is appropriate. In intuitive terms, we can think of low resistance distance between two vertices as indicating that they are “close” in some sense. Note that the resistance distance r(i, j) is affected not just by the ordinary graph theoretic distance between vertices i and j, but also by the number of paths between i and j, as illustrated by Fig. 4.7. Let G be a weighted connected graph on n vertices. For any pair of vertices i, j, let P (i, j) denote the set of all paths from i to j in G, and let δ(i, j) = P 1 min{ e∈P w(e) | P ∈ P (i, j)}. Observe that when the graph is unweighted, δ(i, j) is just the usual graph-theoretic distance between vertices i and j. Note also that this definition of δ(i, j) generalizes the earlier definition given in Corollary 4.5.4 (ii), which only applied to weighted trees. We intend to establish a relationship between δ(i, j) and r(i, j); the following lemma will assist us in doing so.

Lemma 4.8.2. Let G be a weighted graph on n vertices with Laplacian matrix L, ﬁx indices i, j, and let θ > 0 be given. Form G˜ from G by adding θ to the weight of the edge between vertices i and j, so that the Laplacian matrix for G˜ is > L˜ = L + θ(ei − ej)(ei − ej) . Then,

˜# # θ # > # L = L − > # L (ei − ej)(ei − ej) L . 1 + θ(ei − ej) L (ei − ej)

Proof. Use Corollary 4.3.5 and the Sherman–Morrison formula; see the book Horn– Johnson [9, Sect. 0.7.4].

Proposition 4.8.3. Let G be a weighted graph on n vertices. For each i, j = 1, . . . , n, r(i, j) ≤ δ(i, j). 156 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

Proof. It follows from Lemma 4.8.2 that if we delete any edge of G (while retaining the connectivity of the resulting graph), we can only increase the resistance distance between any two vertices. Fix vertices i and j, consider a path Pˆ yielding the minimum value of δ(i, j), let T be a spanning (weighted) tree of G for which the unique path from i to j is Pˆ, and letr ˜(i, j) be the resistance distance from i to j in T . Evidently, r(i, j) ≤ r˜(i, j). Recall thatr ˜(i, j) is the diagonal entry corresponding to vertex i in the bottleneck matrix for T based at vertex j. P Referring to Theorem 4.5.1, we see thatr ˜(i, j) = e∈Pˆ 1/w(e) = δ(i, j). Hence, r(i, j) ≤ r˜(i, j) = δ(i, j) as desired.

Example 4.8.4. We revisit the unweighted graph (K2 ∪ K1) ∨ K2 as depicted in Fig. 4.6. Denoting its resistance distance matrix by R and its graph-theoretic 5 distance matrix by ∆ = δ(i, j) i,j=1, we ﬁnd that

 2 19 19 3  0 5 40 40 5  0 1 1 1 1   2 0 19 19 3   5 40 40 5   1 0 1 1 1   19 19 1 7    R =  0  and ∆ =  1 1 0 1 2  .  40 40 2 8     19 19 1 7  1 1 1 0 2  0     40 40 2 8  1 1 2 2 0 3 3 7 7 5 5 8 8 0

Evidently, R ≤ ∆ entrywise, illustrating the conclusion of Proposition 4.8.3. Next we compute the sum of resistance distances in a weighted graph. Theorem 4.8.5. Let G be a connected weighted graph with Laplacian matrix L. P # Then, 1≤i

n n n n X 1 X X 1 X X r(i, j) = r(i, j) = l# + l# − 2l# . 2 2 i,i j,j i,j 1≤i

Since the row sums of L# are all zero, it follows that

n n n X X # # # X # # # (li,i + lj,j − 2li,j) = nli,i + trace (L ) = 2n trace (L ). i=1 j=1 i=1

The conclusion follows. P The quantity 1≤i

Example 4.8.6. Consider the unweighted path Pn on n vertices, with vertex j adjacent to vertex j + 1 for each j = 1, . . . , n − 1. Denote the corresponding Laplacian matrix by L. Referring to Example 4.5.5, we ﬁnd that

j(j − 1)(2j − 1) + (n − j)(n − j + 1)(2n − 2j + 1) l# = , j,j 6n2 for j = 1, . . . , n. Consequently, we have

n X j(j − 1)(2j − 1) + (n − j)(n − j + 1)(2n − 2j + 1) K(Pn) = n 6n2 j=1 n 1 X = j(j − 1)(2j − 1). 3n j=1

Pn Pn 2 Using the standard sums j=1 j = n(n + 1)/2, j=1 j = n(n + 1)(2n + 1)/6, Pn 3 2 2 2 and j=1 j = n (n + 1) /4, it now follows that K(Pn) = n(n − 1)/6. Next, we consider the eﬀect on the Kirchhoﬀ index of adding the (unweighted) edge between vertices 1 and n, to form the n-cycle, Cn. Letting L˜ denote the Laplacian matrix for Cn, from Lemma 4.8.2 we have

˜# # 1 # > # L = L − > # L (e1 − en)(e1 − en) L . 1 + (e1 − en) L (e1 − en)

Since the trace of a rank-1 matrix uv> (for some pair of vectors u, v ∈ Rn) is equal to v>u, it follows that

n > # # K(Cn) = K(Pn) − > # (e1 − en) L L (e1 − en). 1 + (e1 − en) L (e1 − en)

Referring to Example 4.5.5, we have

> # # # # (e1 − en) L (e1 − en) = l1,1 + ln,n − 2l1,n n(n − 1)(2n − 1) n(n − 1)(2n − 1) n(n − 1)(2n − 1 n(n − 1) = + − 2 − 6n2 6n2 6n2 2n = n − 1.

> # # # Hence, 1 + (e1 − en) L (e1 − en) = n. Using the formulas for lj,1 and lj,n in # # Example 4.5.5 and simplifying, we obtain the fact that lj,1 − lj,n = (n + 1)/2 − j, > # # for each j = 1, . . . , n. Consequently, we find that (e1 − en) L L (e1 − en) = Pn 2 2 j=1((n + 1)/2 − j) = n(n − 1)/12. 2 Assembling the above observations, we find that while K(Pn) = n(n − 1)/6, 2 K(Cn) = n(n − 1)/12. Thus, the addition of the single edge 1 ∼ n reduces the Kirchhoff index to half of its original value. 158 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

Next we present a lower bound on the Kirchhoﬀ index; we begin with the following helpful result. n Lemma 4.8.7. Let x, y ∈ R be two positive vectors such that x1 ≥ · · · ≥ xn Pk Pk and y1 ≥ · · · ≥ yn, and with x majorizing y, i.e., j=1 xj ≥ j=1 yj, for each Pn Pn Pn Pn k = 1, . . . , n − 1, and j=1 xj = j=1 yj. Then, j=1 1/xj ≥ j=1 1/yj. Pk Pk Proof. Set a0 = 0, b0 = 0, and let ak = j=1 xj, and bk = j=1 yj, for k = 1, . . . , n. From the hypothesis note that ak ≥ bk, k = 1, . . . , n − 1, and an = bn. For each j = 1, . . . , n, let cj = −1/(xjyj), and observe that cj ≥ cj+1. Pn Pn We have j=1(1/xj − 1/yj) = j=1 cj(xj − yj). Since xj = aj − aj−1 and yj = bj − bj−1 for j = 1, . . . , n, we ﬁnd that

n n X 1 1 X − = cj[(aj − aj−1) − (bj − bj−1)] x y j=1 j j j=1

n n X X = cj(aj − bj) − cj(aj−1 − bj−1). j=1 j=1

Recalling that an − bn = 0, a0 − b0 = 0, we ﬁnd that

n n−1 n X 1 1 X X − = cj(aj − bj) − cj(aj−1 − bj−1) x y j=1 j j j=1 j=2

n−1 X = (cj − cj+1)(aj − bj) ≥ 0. j=1

Theorem 4.8.8. Let G be a connected weighted graph on vertices 1, . . . , n, and for each j = 1, . . . , n, let dj be the sum of the weights of the edges that are incident Pn with vertex j. For any t > 0, we have K(G) ≥ n j=1 1/(dj + t) − 1/t.

Proof. Let L be the Laplacian matrix for G, and label its eigenvalues as 0 = λ1 < λ2 ≤ · · · ≤ λn. Consider the matrix L+tJ, which has eigenvalues nt (with 1 as an eigenvector), and λ2, . . . , λn (with the same eigenspaces as L); note also that the diagonal entries of L+tJ are dj +t, j = 1, . . . , n. From Schur’s theorem, the vector of eigenvalues of L + tJ majorises the vector of diagonal entries when both vectors are written in nonincreasing order; see Horn–Johnson [9, Sect. 4.3]. Appealing to Pn Pn Lemma 4.8.7, we ﬁnd that 1/(nt) + j=2 1/λj ≥ j=1 1/(dj + t). The conclusion now follows. Corollary 4.8.9. Let G be a connected weighted graph on n vertices, and suppose that the sum of all edge weights is m. Then K(G) ≥ n(n − 1)2/(2m). Proof. We adopt the notation of Theorem 4.8.8. Fix a t > 0. It is straightforward Pn 2 Pn 2 to show that j=1 1/(dj + t) ≥ n /( j=1(dj + t)) = n /(2m + nt). Applying 4.8. Resistance Distance 159

Theorem 4.8.8 and substituting in t = 2m/(n(n−1)) yields the desired inequality.

Suppose that G is a connected unweighted graph on n vertices with m edges. As a variant of the Kirchhoff index for G, we next consider the minimum Kirchhoff index for G, defined as K(G) = inf{K(G)}, where the infimum is taken over all nonnegative weightings of G such that the sum of the edge weights is m. In fact, it turns out that the infimum is always attained as a minimum, since the eigenvalues of the Laplacian matrix of any connected weighted graph are continuous functions of the edge weights. Our next result is intuitively appealing. Theorem 4.8.10. Suppose that G is a connected graph on n vertices. Then, K(G) ≥ n − 1, with equality holding if and only if G = Kn. Proof. Denote the number of edges by m. From Corollary 4.8.9 it follows that K(G) ≥ n(n−1)2/(2m) and, since 2m ≤ n(n−1), we find easily that K(G) ≥ n−1. If K(G) = n − 1, then necessarily 2m = n(n − 1), i.e., G = Kn. Conversely, if G = Kn we find that K(G) = n − 1 ≥ K(G) ≥ n − 1, so that K(G) = n − 1. We have the following explicit formula for trees. Theorem 4.8.11. Let T be a tree on vertices 1, . . . , n, and fix a vertex i. For each edge e of T , let x(e) = |βi(e)|(n − |βi(e)|). Then, 2 1 X p K(T ) = x(e) . (4.6) n − 1 e∈T

In particular, K(T ) ≥ (n − 1)2, with equality if and only if T is a star. Proof. For any weighting w of the tree T , from Corollary 4.5.4 we find that the corresponding Kirchhoff index is equal to P x(e)/w(e). Thus, in order to find P e∈T P K(T ), we need to minimize e∈T x(e)/w(e) subject to the constraint e∈T w(e) = n − 1. Applying Lagrange multipliers, we find that for a minimizing weighting w of T , it must be the case that x(e)/w(e)2 takes on a common value for every edge. P Invoking the constraint e∈T w(e) = n − 1, it now follows that, for each edge e, p P p w(e) = (n − 1) x(e)/ e∈T x(e), yielding (4.6). From the fact that |βi(e)|(n − |βi(e)|) = x(e) ≥ n − 1 for any edge e of T , we 2 find that K(T ) ≥ (n−1) . Further, equality holds if and only if |βi(e)|(n−|βi(e)|) = n−1 for every edge e; the latter is readily seen to hold if and only if T is a star.

Example 4.8.12. Consider the unweighted path on n vertices, Pn, and label the vertices so that vertex k is adjacent to vertex k + 1, for k = 1, . . . , n − 1. Letting e denote the edge between vertex k and vertex k + 1, we ﬁnd readily that x(e) = k(n − k). Consequently, by Theorem 4.8.11, we have

n−1 !2 1 X p K(Pn) = k(n − k) . n − 1 k=1 160 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

u u e1 e2

u u u u u e3 e4 e5 e6 √ √ √6 6 √ ≈ w(ej) = 4 6+ 10+ 12 0.8949, j = 1, 2, 3, 6

√ √ 6√ 10 √ ≈ w(e4) = 4 6+ 10+ 12 1.1552

√ √ 6√ 12 √ ≈ w(e5) = 4 6+ 10+ 12 1.2655

Figure 4.8: Optimally weighted tree for Example 4.8.13

Pn−1 p 2 R 1 p Note that, as n → ∞, k=1 k(n − k) /n → 0 t(1 − t)dt. Using the sub- 2 R 1 p R 2π 2 2 stitution t = sin (θ/4) yields 0 t(1 − t)dt = 0 sin (θ/4) cos (θ/4)dθ /2. Fi- nally, using the trigonometric identity 1− cos(θ) = 8 sin2(θ/4) cos2(θ/4), it follows R 1 p 3 2 that 0 t(1 − t)dt = π/8. We thus deduce that K(Pn) is asymptotic to n π /64, as n → ∞. Example 4.8.13. Here we revisit the tree in Example 4.5.2 and ﬁnd its optimal weighting yielding the minimum Kirchhoﬀ index; see Fig. 4.8. Observe that for each of the pendent edges ej, j = 1, 2, 3, 6, we have x(ej) = 6, while x(e4) = 10 and x(e5) = 12. The edge weights now follow from Theorem 4.8.11.

4.9 Computational Considerations

The results in the preceding sections suggest that the group inverse of the Lapla- cian matrix for a connected graph carries some interesting information about the graph. The question then arises: how might one compute that group inverse? In this section, we discuss that issue. Note that both Corollary 4.3.5 and Theorem 4.4.1 give closed-form formulas for the group inverse of the Laplacian matrix of a connected graph. Both results involve the computation of a matrix inverse – the former expression uses the inverse 1 of L + n J (for a graph on n vertices with Laplacian L) and the latter requires the computation of a bottleneck matrix; in either case, these can be computed using the LU decomposition (see Golub–Van-Loan [7, Sect. 3.2]) in roughly 4n3/3 floating point operations (additions or multiplications), or flops. While flop counts are not the only factor determining the efficiency of an algorithm, they can be 4.9. Computational Considerations 161 indicative of the overall workload, hence our interest in them. Remark 4.9.1. Observe that the LU approach above does not take advantage of the fact that the Laplacian matrix of a connected graph is positive semidefinite. The following strategy uses some of the extra structure of the Laplacian matrix. Recall that any symmetric positive definite matrix S has a Cholesky decomposition. That is, there is a lower triangular matrix N so that S = NN >. Here is a sketch of how one can establish the existence of such an N. Write s s> S = 1,1 2,1 . s2,1 S2,2

R Rn−1 2 Since S is positive definite, we find that for any x1 ∈ , x2 ∈ , x1s1,1 + > > > 2x1s2,1x2 + x2 S2,2x2 > 0. Taking x1 = −s2,1x2/s1,1, we find that for any x2 ∈ n−1 > 1 > 1 > R , x (S2,2 − s2,1s )x2 > 0. Hence, S2,2 − s2,1s is positive defi- 2 s1,1 2,1 s1,1 2,1 > nite. Applying an induction step, let Nˆ be lower triangular with NˆNˆ = S2,2 − 1 > s2,1s . Now take s1,1 2,1 " √ > # s1,1 0 N = 1 . √ s2,1 Nˆ s1,1 We refer the interested reader to Golub–Van-Loan [7, Sect. 4.2.3] for more details on the Cholesky decomposition. Referring to Theorem 4.4.1, observe that for the Laplacian matrix L of a connected graph on n vertices, the principal submatrix L{n} formed by deleting the last row and column of L is a positive definite matrix. Computing a Cholesky > decomposition of L{n} as L{n} = NN , we can then find the bottleneck matrix −1 −> −1 # B = L{n} via B = N N , and apply Theorem 4.4.1 to compute L . The Cholesky decomposition of an n × n symmetric positive definite matrix can be computed in roughly n3/3 flops, while the inversion of the triangular Cholesky factor N can be computed in another n3/3 flops. The product N −>N −1 can also be computed in about n3/3 flops, and consequently, this approach allows us to compute L# in about n3 flops. Remark 4.9.2. Suppose that we have a weighted tree on n vertices. A straightforward proof by induction shows that there is a labeling of the vertices of T with the numbers 1, . . . , n and the edges of T with the labels e(1), . . . , e(n − 1) such that (i) for each j = 1, . . . , n − 1, the vertex j is incident with the edge e(j), and (ii) for each j = 1, . . . , n − 1, the endpoint of e(j) distinct from j, say kj, is such that kj > j. With this labeling in place, we can construct a square matrix N = (ni,j) of order n − 1 as follows: p  w(e(j)) if i = j,   p ni,j = − w(e(j)) if i = kj,  0 if i =6 j, kj. 162 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

2u 3 u e2 e3

u u u u u 1e1 5e5 6e6 7e4 4

Figure 4.9: The labeled tree for Example 4.9.3

It is readily seen that if we set " # N H = , −1>N then HH> is the Laplacian matrix for T . Further, the special labeling of vertices and edges ensures that N is lower triangular, so that it is in fact the Cholesky factor referred to in Remark 4.9.1. Thus, for the special case of a tree, the Cholesky factor N can be constructed from the labeling of the vertices and edges; this strategy then allows for the computation of the group inverse of the corresponding Laplacian matrix in roughly 2n3/3 ﬂops.

Example 4.9.3. Again, we revisit the tree in Fig. 4.5 that was discussed in Exam- ple 4.5.2. We apply the labeling of vertices and edges discussed in Remark 4.9.2, and produce the labeled tree in Fig 4.9. This in turn produces the oriented incidence matrix

 1 0 0 0 0 0   0 1 0 0 0 0     0 0 1 0 0 0  " #   N H =  0 0 0 1 0 0  ≡ .   −1>N  −1 0 0 0 1 0     0 −1 0 0 −1 1  0 0 −1 −1 0 −1

A computation now yields

 1 0 0 0 0 0   0 1 0 0 0 0    −1  0 0 1 0 0 0  N =   ,  0 0 0 1 0 0     1 0 0 0 1 0  1 1 0 0 1 1 4.9. Computational Considerations 163

1u 2 u JJ J J uJ u u 3 5 4

Figure 4.10: The star K1,4 so that the bottleneck matrix based at vertex 7 is given by  3 1 0 0 2 1   1 2 0 0 1 1    −> −1  0 0 1 0 0 0  B = N N =   .  0 0 0 1 0 0     2 1 0 0 2 1  1 1 0 0 1 1 Denoting the Laplacian matrix for this labeled tree by L and applying Theo- rem 4.4.1, it now follows that  73 −11 −32 −32 31 −4 −25   −11 52 −18 −18 −4 10 −11       −32 −18 59 10 −25 −11 17  # 1   L =  −32 −18 10 59 −25 −11 17  . 49    31 −4 −25 −25 38 3 −18       −4 10 −11 −11 3 17 −4  −25 −11 17 17 −18 −4 24 Remark 4.9.4. Note that for connected graphs with not very many edges m, say n− 1 plus a constant, the following strategy may be effective: (i) find a spanning tree; a standard algorithm such as Kruskal’s algorithm (see Bondy–Murty [3, Sect. 2.5]), does this in n2 steps; (ii) compute the group inverse of the corresponding Laplacian matrix via Remark 4.9.2 (at a cost of 2n3/3 flops); (iii) update the Laplacian matrix using Lemma 4.8.2 by adding in the remaining m − n + 1 weighted edges, one at a time. Observe that each update step costs roughly n2 flops. Example 4.9.5. We now illustrate this updating technique. We begin with the star K1,4 from Fig. 4.10 The group inverse for the corresponding Laplacian matrix L1 is  19 −6 −6 −6 −1   −6 19 −6 −6 −1  # 1   L =  −6 −6 19 −6 −1  . 1 25    −6 −6 −6 19 −1  −1 −1 −1 −1 4 164 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

1u 2 u JJ J J J J J u J uJJ u 3 5 4

Figure 4.11: The graph after the three edge additions

Next, we add the edge 1 ∼ 2, and denote the corresponding Laplacian matrix # > by L2. Since L1 (e1 −e2) = 1 −1 0 0 0 , we now ﬁnd from Lemma 4.8.2  32 7 6 6 1  75 75 − 25 − 25 − 25  7 32 6 6 1   75 75 − 25 − 25 − 25    #  − 6 − 6 19 − 6 − 1  L2 =  25 25 25 25 25  .    − 6 − 6 − 6 19 − 1   25 25 25 25 25  1 1 1 1 4 − 25 − 25 − 25 − 25 25

Now add the edge 1 ∼ 3. Denoting the new Laplacian matrix by L3 and using # > the fact that L2 (e1 − e3) = 2/3 1/3 −1 0 0 , Lemma 4.8.2 yields  13 1 1 6 1  50 100 100 − 25 − 25  1 77 23 6 1   100 200 − 200 − 25 − 25    #  1 − 23 77 − 6 − 1  L3 =  100 200 200 25 25  .    − 6 − 6 − 6 19 − 1   25 25 25 25 25  1 1 1 1 4 − 25 − 25 − 25 − 25 25

Finally, we add the edge 2 ∼ 4 to get the graph depicted in Fig 4.11. Take L4 as the # > corresponding Laplacian matrix. Since L3 (e2−e4)= 1/4 5/8 1/8 −1 0 , we ﬁnd from Lemma 4.8.2 that  124 26 1 76 1  525 − 525 − 525 − 525 − 25  26 124 76 1 1   − 525 525 − 525 − 525 − 25    #  − 1 − 76 199 − 101 − 1  L4 =  525 525 525 525 25  .    − 76 − 1 − 101 199 − 1   525 525 525 525 25  1 1 1 1 4 − 25 − 25 − 25 − 25 25 One issue of concern in numerical linear algebra is the sensitivity of a computation to small changes in the input parameters. If we have two Laplacian matrices L and L˜ that we believe are “close”, what guarantees do we have that L# and L˜# are close? The follow result provides some insight. 4.9. Computational Considerations 165

Theorem 4.9.6. Suppose that G and G˜ are connected weighted graphs on n vertices, and denote their corresponding Laplacian matrices by L and L˜, respectively. Set F = L − L˜, and suppose that ||F ||2 < α(G). Then,

˜# # ||F ||2 ||L − L ||2 ≤ 2 . α(G) − α(G)||F ||2

# # Proof. First we note that, since ||F ||2 < α(G), we have ||FL ||2 ≤ ||F ||2||L ||2 = # ||F ||2/α(G) < 1. Hence, I − FL is invertible. It is straightforward to determine that L˜# = L#(I − FL#)−1, and hence we ﬁnd that L˜# − L# = L#FL#(I − FL#)−1. Taking norms now yields

# 2 ˜# # # # # −1 ||L ||2||F ||2 ||L − L ||2 ≤ ||L FL ||2||(I − FL ) ||2 ≤ # . 1 − ||L ||2||F ||2

# Using ||L ||2 = 1/α(G) and simplifying, we get the desired inequality.

Example 4.9.7. Consider the unweighted star on n vertices, with center vertex n. We have

" # " n+1 1 # I −1 # I − n2 J − n2 1 L = > and L = 1 > n−1 . −1 n − 1 − n2 1 n2

Now perturb the weight of the edge between vertices 1 and 2 so that it has weight > 0, and let L˜ denote the corresponding Laplacian matrix. In the notation above, > # F = L−L˜ = −(e1 −e2)(e1 −e2) , so that ||F ||2 = 2. Since L (e1 −e2) = e1 −e2, # ˜# > it follows from Lemma 4.8.2 that L − L = 1+2 (e1 − e2)(e1 − e2) . Thus # # kL˜ − L ||2 = 2/(1 + 2). The algebraic connectivity of the star is 1, and so the upper bound of the preceding result is given by 2/(1 − ). Observe that, for this example, L# is not very sensitive to changes in the weight of the edge between vertices 1 and 2.

Example 4.9.8. Consider Pn with vertices labeled in increasing order from one pendent vertex to the other. Let L be the corresponding Laplacian matrix, and > consider L˜ = L + (e1 − en)(e1 − en) , where > 0. From Lemma 4.8.2, we have

# ˜# # > # L − L = > # L (e1 − en)(e1 − en) L . 1 + (e1 − en) L (e1 − en)

# # It’s straightforward to determine that lj,1 − lj,n = (n + 1)/2 − j, for j = 1, . . . , n. > # > # # It follows that (e1 − en) L (e1 − en) = n − 1 and (e1 − en) L L (e1 − en) = 2 # # 2 n(n − 1)/12. We deduce that kL − L˜ k2 = n(n − 1)/(12(1 + (n − 1))). In particular, if n is large, L# may be quite sensitive to small changes in the weight of 2 2 the edge between vertices 1 and n. Recall that α(Pn) = 2(1 − cos(π/n)) ≈ π /n . 166 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

In Bozzo–Franceschet [4], the authors present a strategy for approximating the group inverse of the Laplacian matrix for a weighted graph. The motivation for doing so stems from the fact that some graphs, such as those arising in biological or social networks, are so large that computing the group inverse of the Laplacian is infeasible due to time and space limitations. Nevertheless, the group inverse of the Laplacian matrix is of interest in analyzing these graphs, as it furnishes measures of centrality that are based on the analogy with resistive electrical networks. Consequently, an approximation to the desired group inverse is produced along the following lines. Suppose that we have a connected weighted graph G on n vertices with Lapla- cian matrix L. Denote the eigenvalues of L by 0 ≡ λ1 < λ2 ≤ · · · ≤ λn, and let Q > be an orthogonal matrix of eigenvectors with L = Q diag( 0 λ2 ··· λn )Q . # > Then, by Theorem 4.3.3 we have L = Q diag( 0 1/λ2 ··· 1/λn )Q or, # Pn 1 > > equivalently, L = Qeje Q . j=2 λj j This observation motivates the following deﬁnition: for each k = 2, . . . , n, let

k (k) X 1 > > T = Qeje Q . (4.7) λ j j=2 j

Bozzo–Franceschet [4] refers to T (k) as the k-th cutoff approximation for L#; evidently, T (n) is equal to L#. The computational advantage of approximating L# by T (k) is that only k −1 eigenpairs (i.e., eigenvalues and corresponding eigenvectors) need to be computed; if k is substantially smaller than n, the time and storage needed to compute T (k) are substantially reduced compared to the computation of L#. In order to compute these k − 1 eigenpairs, we note that for large sparse symmetric matrices (such as the Laplacian matrix of a graph on n vertices where n the number of edges is small relative to 2 ) the Lanczos method, or one of its variants, is a natural tool to consider. This is because for symmetric matrices, the Lanczos method (an iterative method) is well suited for computing the extreme eigenvalues and corresponding eigenvectors. We refer the interested reader to Golub–Van-Loan [7, Ch. 9], which provides a thorough discussion of Lanczos methods. It is straightforward to see that for each 2 ≤ k ≤ n − 1, (k) 0 1 ··· 1 0 ··· 0 > T = Q diag λ2 λk Q . Hence, h i 1 || # − (k)|| 0 ··· 0 1 ··· 1 > L T 2 = Q diag λk+1 λn Q = . 2 λk+1 # Clearly, ||L ||2 = 1/λ2 and so we find that the relative error (in the 2-norm) of the k-th cutoff approximation is given by

# (k) ||L − T ||2 λ2 # = . ||L ||2 λk+1 4.9. Computational Considerations 167

Observe that the relative error is nonincreasing in k. Example 4.9.9. Again, we consider the unweighted star on n vertices with center vertex n. Recall that the nonzero eigenvalues for the corresponding Laplacian matrix L are λj = 1 for j = 2, . . . , n − 1, and λn = n. The eigenspace for L corresponding to the eigenvalue n is spanned by the unit vector 1 −1n−1 qn = p , n(n − 1) n − 1 while the eigenspace for L corresponding to the eigenvalue 1 is the orthogonal complement of the subspace spanned by 1 and qn. The fact that the eigenspace corresponding to the eigenvalue 1 has dimension n − 2 means that, for each 2 ≤ k ≤ n − 2, there are many diﬀerent possible choices of T (k) given by (4.7): indeed, for any collection of k − 1 orthonormal vectors q2, . . . , qk in the 1-eigenspace, we (k) Pk > # may form T = j=2 qjqj to produce an approximation to L . Observe that the relative error for any such T (k) is 1. On the other hand, T (n−1) is uniquely deﬁned and is given by " # I − 1 J 0 T (n−1) = n−1 . 0> 0

This can be deduced from the fact that, for any orthonormal basis q2, . . . , qn−1 of 1 Pn > the eigenspace corresponding to the eigenvalue 1, we have I = n J + j=2 qjqj , Pn−1 > 1 > so that j=2 qjqj = I − n J − qnqn .

Example 4.9.10. Here, we revisit the unweighted path Pn. Referring to Exam- ple 4.6.4, we ﬁnd that for each j = 2, . . . , n, the Laplacian matrix L for the path has λj = 2 (1 − cos(π(j − 1)/n)) as an eigenvalue, with

r > 2 h jπ 3jπ 5jπ (2n−1)jπ i qj = cos( ) cos( ) cos( ) ··· cos n 2n 2n 2n 2n as a corresponding unit eigenvector. The fact that qj is a unit vector can be veriﬁed by observing that n n (2k−1)jπ X (2k − 1)jπ X cos n + 1 n cos2 = = . 2n 2 2 k=1 k=1

In contrast to Example 4.9.9, each eigenspace is one-dimensional, and so T (k) of (4.7) is uniquely determined for each k = 2, . . . , n−1. Here we ﬁnd that the relative error for the k-th cutoﬀ approximation T (k) is equal to

π 2(1 − cos( n )) π(k−1) . 2(1 − cos( n )) 168 Chapter 4. The Group Inverse of the Laplacian Matrix of a Graph

Observe that for ﬁxed k, this relative error is asymptotic to 1/(k − 1)2 as n → ∞. Next we try to get a sense of how well particular entries of L# (speciﬁcally, diagonal ones) are approximated by the corresponding entry of T (k). It is straightforward to see that, for each 1 ≤ m ≤ n, the (m, m) entry of T (k) is given by k 2 (2m−1)(j−1)π 1 X cos 2n . (4.8) n π(j−1) j=2 1 − cos n

For ﬁxed m and k, we ﬁnd that (4.8) is asymptotic to

k k−1 1 X 1 2n X 1 = . n π2(j−1)2 π2 j2 j=2 2n2 j=1

Recall that, from Example 4.5.5, the (m, m) diagonal entry of L# is given by m(m − 1)(2m − 1) + (n − m)(n − m + 1)(2n − 2m + 1)/(6n2) which, for fixed # (k) m, is asymptotic to n/3 as n → ∞. Thus, for fixed m and k, Lm,m − Tm,m is 2n π2 Pk−1 1 2n P∞ 1 asymptotic to π2 6 − j=1 j2 = π2 j=k j2 , the equality following from the P∞ 2 2 fact that j=1 1/j = π /6; see Aigner–Ziegler [2, Sect. 9, Ch. 1] for several different proofs of that identity.

4.10 Closing Remarks

Just as the Laplacian matrix L of a connected graph carries information about that graph (explicitly via its entries, but also implicitly, as evidenced by the Matrix Tree Theorem), we have seen that L# also carries graph-theoretic information. In particular, the use of the All Minors Matrix Tree Theorem yields a combinatorial interpretation of the entries of a bottleneck matrix, which is in turn the key in- gredient in the formula for L#; indeed the connection between the entries in L# and the structure of the graph is especially clear when the graph in question is a tree. We have also seen how L# provides bounds on the algebraic connectivity of a graph, and how, via a parallel between weighted graphs and resistive electrical networks, L# is naturally associated with two notions of “closeness”: the resistance distance between a given pair of vertices, and the Kirchhoﬀ index, which yields a global measure for the entire network. Because of the mathematically appealing structure of the Laplacian matrix of a connected graph – i.e., it is a symmetric, positive semideﬁnite, singular M-matrix whose null space is known – there are several options available for computing L# numerically. As it happens, the group inverse of a singular and irreducible M-matrix arises in a variety of settings, including the analysis of Markov chains, perturbation results for Perron eigenvalues and eigenvectors, and matrix population models. For readers seeking further details, an array of properties, results, and applications 4.10. Closing Remarks 169 of group inverses for singular, irreducible M-matrices is presented in Kirkland– Neumann [11]. Bibliography

[1] N. Abreu, “Old and new results on algebraic connectivity of graphs”, Linear Algebra and its Applications 423 (2007), 53–73. [2] M. Aigner and G. Ziegler, “Proofs from the Book”, 5th Edition. Springer, Berlin, 2014. [3] J. Bondy and U. Murty, “Graph theory with applications”, American Elsevier Publishing Co., New York, 1976. [4] E. Bozzo and M. Franceschet, “Approximations of the generalized inverse of the graph Laplacian matrix”, Internet Mathematics 8 (2012), 456–481. [5] R. Brualdi and H. Ryser, “Combinatorial matrix theory”, Cambridge Univer- sity Press, Cambridge, 1991. [6] S. Chaiken, “A combinatorial proof of the all minors matrix tree theorem”, SIAM Journal on Algebraic and Discrete Methods 3 (1982), 319–329. [7] G. Golub and C. Van Loan, “Matrix computations”, 3rd Edition. Johns Hop- kins University Press, Baltimore, 1996. [8] P. Halmos, “Finite dimensional vector spaces”, 2nd Edition. Springer-Verlag, New York, 1958. [9] R. Horn and C. Johnson, “Matrix analysis”, 2nd Edition. Cambridge Univer- sity Press, Cambridge, 2012. [10] S. Kirkland, “Algebraic connectivity”, in “Handbook of Linear Algebra”, 2nd Edition, L. Hogben (Ed.). CRC Press, Boca Raton, 48–1 to 48–14, 2014. [11] S. Kirkland and M. Neumann, “Group inverses of M-matrices and their applications”, CRC Press, Boca Raton, 2013. [12] N. Mahadev and U. Peled, “Threshold graphs and related topics”, Annals of Discrete Mathematics 56 (1995), 542. [13] J. Molitierno, “Applications of combinatorial matrix theory to Laplacian matrices of graphs”, CRC Press, Boca Raton, 2012. [14] E. Seneta, “Non-negative matrices and Markov chains”, 2nd Edition. Springer-Verlag, New York, 1981. Chapter 5

Boundary Value Problems on Finite Networks by Angeles´ Carmona

5.1 Introduction

This chapter is motivated by a well-known matrix problem; specifically, the M- matrix inverse problem, as we will see in the next section. Our approach differs from others because the tools we use come from discrete potential theory, in which we have been working for a long period, trying to emulate as much as possible the continuous case. This chapter introduces this way of approximating a problem typical of matrix theory and offers an overview of the potential power of introducing new approaches in this field. First, we aim at introducing the basic terminology and results on discrete vector calculus on finite networks. After defining the tangent space at each vertex of a network, we introduce the basic difference operators that substitute the usual differential operators. Specifically, we define the derivative, gradient, divergence, curl, and Laplacian operators, or, more generally, the Schödingeroperator. Moreover, we prove that the above-defined operators satisfy properties that are analogues to those satisfied by their continuous counterparts. We must note that as the space is discrete all these objects are nothing else but vectors and matrices fields. The next step consists of studying the basic terminology and results on self- adjoint boundary value problems on finite networks. First we define the discrete analogue of a manifold with a boundary, which includes the concept of an outer normal field. Then, we prove the Green Identities in order to establish the vari- 174 Chapter 5. Boundary Value Problems on Finite Networks ational formulation of boundary value problems. Moreover, we prove the discrete version of the Dirichlet Principle. Once we have studied and defined the basic operators and their relations, we aim at analyzing the fundamental properties of the positive semidefinite Schrö- dinger operator on networks, both with boundaries and without. We prove the properties of monotonicity and the minimum principle, which allows us to define the Green and Poisson operators associated with the Schrödingeroperators. In the case of a network with a boundary or when F = V but q =6 qσ, the Green operator can be thought of as the inverse matrix of Lq, whereas when F = V and q = qσ it can be seen as the group inverse. After analyzing the properties of and relationships between the two operators we define the associated kernels as well as the Dirichlet-to-Robin map. At the end of the chapter we present an application of these results to matrix theory. First, we introduce the concept of effective resistance and the Kirchhoff index with respect to a parameter and a weight, and we find the relation between effective resistances and Green functions. These definitions have allowed us to extend the M-matrix inverse problem to non-diagonally dominant matrices and to give a characterization of when a singular symmetric and irreducible M-matrix has a group inverse that is also an M-matrix.

5.2 The M-Matrix Inverse Problem

The matrices1 that can be expressed as L = kI−A, where k > 0 and A ≥ 0, appear in relation to systems of equations or eigenvalue problems in a wide variety of areas including finite difference methods for solving partial differential equations, input- output production and growth models in economics, and Markov processes in probability and statistics. Of course, the combinatorial community can recognize, within this group of matrices, the combinatorial Laplacian of a k-regular graph where A is the adjacency matrix. If k is at least the spectral radius of A, then L is called an M-matrix. M- matrices satisfy monotonicity properties that are the discrete counterpart of the minimum principle, and this makes them suitable for the resolution of large sparse systems of linear equations by iterative methods. In fact, the properties of M- matrices are the discrete analogue of the properties of second-order elliptic operators. This chapter is motivated by this analogy. A well-known property of an irreducible non-singular M-matrix is that its inverse is positive; see Berman–Plemmons [11]. It is known that every irreducible and singular M-matrix has a generalized inverse which is nonnegative, but this is not always true of every generalized inverse. For instance, the group inverse of the combinatorial Laplacian of a path always has some negative off-diagonal entries.

1At the end of this chapter there is a glossary including the main terminology related to matrices. 5.2. The M-Matrix Inverse Problem 175

The diﬃculty of characterizing all nonnegative matrices whose inverses are M-matrices has led to the study of the general properties of inverse M-matrices and to the identiﬁcation of particular classes of such matrices. Let us begin with some notation and bibliographic revision. Let cij ≥ 0, 1 ≤ i < j ≤ n, and the symmetric matrix which we always assume irreducible   0 c12 ··· c1n−1 c1n  c12 0 ··· c2n−1 c2n     . . . . .  A =  ......  ,    c1n−1 c2n−1 ··· 0 cn−1n  c1n c2n−1 ··· cn−1n 0 and consider the Z-matrix   d1 −c12 · · · −c1n−1 −c1n  −c12 d2 · · · −c2n−1 −c2n     . . . . .  M =  ......  = D − A.    −c1n−1 −c2n−1 ··· dn−1 −cn−1n  −c1n −c2n−1 · · · −cn−1n dn

The diﬀerent problems that we can raise in this framework are the following: (i) Characterization of symmetric M-matrices: for which particular values of d1, . . . , dn > 0 is M an M-matrix? (ii) Inverse M-matrix problems, non-singular case: when M is invertible (a Stieltjes matrix), then M−1 > 0; so, if we consider K > 0 irreducible, symmetric, and invertible, when is K = M−1? (iii) Inverse M-matrix problems, singular case: if M is singular, what can we say about M#? When is M# an M-matrix? There have been many contributions to these problems over the years. For instance, (i) Markham [33] proved that, if K = (amin{i,j}), 0 < a1 < ··· < an, then K−1 is a Jacobi M-matrix; (ii) Mart´ınez–Michon–Mart´ın[34] showed that, if K is a strictly ultrametric matrix, then K−1 is a strictly diagonally dominant (d.d.) Stieltjes matrix; (iii) Fiedler [26] proved that if M is a Stieltjes and d.d. matrix, then M−1 is a resistive inverse matrix; (iv) Chen–Kirkland–Neumann [19] proved that if the group inverse of a singular and d.d. Jacobi M-matrix is an M-matrix, then n ≤ 4; (v) Kirkland–Neumann [28] characterized all weighted trees whose Laplacian has a group inverse which is an M-matrix; and (vi) Carmona et al. also worked in this framework obtaining a generalization of some of the above results; speciﬁcally, they proved that any irreducible Stieltjes matrix is a resistive inverse and that, for any n, there exist singular, symmetric, and Jacobi M-matrices of order n whose group inverse is also an M-matrix; see [6, 8, 9]. 176 Chapter 5. Boundary Value Problems on Finite Networks

The techniques we use are based on the study of potential theory associated with a positive semidefinite Schrödingeroperator on a finite network and can be seen as the discrete counterpart of the potential theory associated with elliptic operators. The connection between finite networks and matrices comes from the following definition. Given cij ≥ 0, 1 ≤ i < j ≤ n, and the irreducible and symmetric matrix   0 c12 ··· c1n−1 c1n  c12 0 ··· c2n−1 c2n     . . . . .  A =  ......     c1n−1 c2n−1 ··· 0 cn−1n  c1n c2n ··· cn−1n 0 we can define a network or weighted graph, G = (V, E, c), whose vertex set is V = {x1, . . . , xn} and where c(xi, xj) = cij, i.e., {xi, xj} ∈ E ⇔ cij > 0. Moreover, Pn κi = κ(xi) = j=1 cij is the degree of xi. In fact, A being irreducible is equivalent to G being connected. Using this definition we can parametrize all the Z-matrices having the same off-diagonal elements as the matrices     κ1 −c12 · · · −c1n−1 −c1n q1 0 ··· 0 0  −c12 κ2 · · · −c2n−1 −c2n   0 q2 ··· 0 0       . . . . .   . . . . .  M =  ......  +  ......  ,      −c1n−1 0 ··· κn−1 −cn−1n   0 0 ··· qn−1 0  −c1n 0 · · · −cn−1n κn 0 0 ··· 0 qn where q1, . . . , qn ∈ R. As we will see, these are the matrices associated with Schrödingeroperators on G, where qi represents the corresponding potential. Clearly, when qi ≥ 0, then M is diagonally dominant and hence positive semidefinite.

5.3 Diﬀerence Operators on Networks

Throughout this chapter, G = (V,E) denotes a simple connected and finite graph without loops, with vertex set V and edge set E. Two different vertices, x, y ∈ V , are called adjacent, which is represented by x ∼ y, if {x, y} ∈ E. In this case, the edge {x, y} is also written as exy and the vertices x and y are called incident with exy. In addition, for any x ∈ V , k(x) denotes the number of vertices adjacent to x. When k(x) = k for all x ∈ V we say that the graph is k-regular. We denote by C(V ) and C(V ×V ) the vector spaces of real functions defined on the sets that appear between brackets. If u ∈ C(V ) and f ∈ C(V × V ), uf denotes 5.3. Difference Operators on Networks 177

c(x,y)

Figure 5.1: Network the function defined for any x, y ∈ V as (uf)(x, y) = u(x)f(x, y). If u ∈ C(V ), the support of u is the set supp(u) = {x ∈ V | u(x) =6 0}. For any u ∈ C(V ) we denote Z P by u dx the value x∈V u(x). V Throughout the paper we make use of the subspace of C(V × V ) given by C(G) = {f ∈ C(V × V ) | f(x, y) = 0 if x 6∼ y}. We call conductance on G a function c ∈ C(G) such that c(x, y) > 0 if and only if x ∼ y. The pair (G, c) is called a network or weighted graph; see Fig. 5.1. In what follows, we consider the network (G, c) fixed and we refer to it simply by G. R The function κ ∈ C(V ) defined as κ(x) = V c(x, y) dy for any x ∈ V is called the (generalized) degree of G. Moreover, the resistance of G is the function r ∈ C(G) defined as r(x, y) = 1/c(x, y), when x ∼ y. Next, we define the tangent space at a vertex of a graph. Given x ∈ V , we call the real vector space of formal linear combinations of the edges incident with x the tangent space at x and we denote it by Tx(G); see Fig. 5.2. So, the set of edges incident with x is a basis of Tx(G), called a coordinate basis of Tx(G), and hence dim Tx(G) = k(x). Note that, in the discrete setting, the dimension of the tangent space varies with each vertex except when the graph is regular. We call any application f : V → ∪x∈V Tx(G) such that f(x) ∈ Tx(G) for each x ∈ V a vector field on G. The support of f is defined as the set supp(f) = {x ∈ V | f(x) =6 0}. The space of vector fields on G is denoted by X (G). If f is a vector field on G, then f is uniquely determined by its components in the coordinate basis. Therefore, we can associate with f the function f ∈ C(G) such P that, for each x ∈ V , f(x) = y∼x f(x, y) exy, and hence X (G) can be identified with C(G). A vector field f is called a flow when its component function satisfies f(x, y) = −f(y, x) for any x, y ∈ V , whereas f is called symmetric when its component 178 Chapter 5. Boundary Value Problems on Finite Networks

Tx(Γ)

Figure 5.2: Tangent space at x function satisfies f(x, y) = f(y, x) for any x, y ∈ V . Given a vector field f ∈ X (G), we consider two vector fields, the symmetric and the antisymmetric fields associated with f, denoted by fs and fa, respectively, that are defined as the fields whose component functions are given, respectively, by

f(x, y) + f(y, x) f(x, y) − f(y, x) f s(x, y) = and f a(x, y) = . 2 2 Observe that f = fs + fa for any f ∈ X (G). If u ∈ C(V ) and f ∈ X (G) has f ∈ C(G) as its component function, the field uf is defined as the field whose component function is uf. If f, g ∈ X (G) and f, g ∈ C(G) are their component functions, the expression hf, gi denotes the function in C(V ) given by X hf, gi(x) = f(x, y)g(x, y)r(x, y) (5.1) y∼x for any x ∈ V . Clearly, for any x ∈ V , h·, ·i(x) determines an inner product on Tx(G). Therefore, on a network we can consider the following inner products on C(V ) and on X (G),

Z 1 Z hu, vi = u v dx, and hf, gi dx (5.2) V 2 V 5.3. Difference Operators on Networks 179 for u, v ∈ C(V ) and f, g ∈ X (G), respectively; the factor 1/2 is due to the fact that each edge is considered twice. Lemma 5.3.1. For f, g ∈ X (G), we have R hf, gi dx = R hfs, gsi dx+R hfa, gai dx. V RV V In particular, if f is symmetric and g is a flow, then V hf, gi dx = 0. We are ready to define the discrete analogues of the fundamental first- and second-order differential operators on Riemannian manifolds, specifically the derivative, gradient, divergence, curl, and Laplacian. The last of these is called a second-order difference operator, whereas the others are generically called first- order difference operators. From now on we suppose the network (G, c) and the associated inner products on C(V ) and X (G), to be fixed. We use the term derivative operator for the linear map d: C(V ) → X (G), P assigning to any u ∈ C(V ) the flow du, given by du(x) = y∼x(u(y) − u(x)) exy, and called the derivative of u. We use the term gradient for the linear map ∇: C(V ) → X (G), assigning to any u ∈ C(V ) the flow ∇u, called the gradient of u, and given by ∇u(x) = P y∼x c(x, y)(u(y) − u(x)) exy. Clearly, it is verified that du = 0, or equivalently ∇u = 0, if and only if u is a constant function. We define the divergence operator as div = −∇∗; that is, the linear map div : X (V ) → C(V ), assigning to any f ∈ X (G) the function div f, called the divergence of f, and determined by the relation Z 1 Z u div f dx = − h∇u, fi dx, (5.3) V 2 V for u ∈ C(V ). Therefore, treating u as constant in the above identity, we obtain R that, for any f ∈ X (G), V div f dx = 0. P a Proposition 5.3.2. For f ∈ X (G) and x ∈ V , div f(x) = y∼x f (x, y).

Proof. For any z ∈ V consider u = εz, the Dirac function at z. Then, from (5.3), 1 R 1 P we get div f(z) = − 2 V h∇εz, fi dx = − 2 x∈V h∇εz, fi(x). For x ∈ V , we get that X X h∇εz, fi(x) = c(x, y) εz(y) − εz(x) f(x, y)r(x, y) = εz(y) − εz(x) f(x, y) y∼x y∈V

X = f(x, z) − εz(x)f(x, y), y∈V y6=z P hence when x =6 z, h∇εz, fi(x) = f(x, z), whereas h∇εz, fi(z) = − y∈V f(z, y). y6=z Therefore,

1 X 1h X X i X a div f(z) = − h∇εz, fi(x) = f(z, y) − f(x, z) = f (z, x). 2 2 x∈V y∈V x∈V x∈V y6=z x6=z 180 Chapter 5. Boundary Value Problems on Finite Networks

We use the term curl for the linear map curl: X (G) → X (G), assigning to any f ∈ X (G) the symmetric vector field curl f, called curl of f, and given by P s curl f(x) = y∼x r(x, y)f (x, y) exy. In the following result we show that the above defined difference operators satisfy properties that match the ones satisfied by their differential analogues. Proposition 5.3.3. curl∗ = curl, div ◦ curl = 0, and curl ◦ ∇ = 0. Proof. Let f, h ∈ X (G), then Z Z X hcurl f, hi dx = hcurl f, hsi dx = r(x, y)2f s(x, y)hs(x, y) V V x,y∈V Z Z = hfs, curl hi dx = hf, curl hi dx. V V The other two equalities follow directly from the definitions of the operators. Now we introduce the fundamental second-order difference operator on C(V ), which is obtained by composition of two first-order operators. Specifically, we consider the endomorphism of C(V ) given by L = −div ◦ ∇, which we call the Laplace operator or combinatorial Laplacian of G. Proposition 5.3.4. For any u ∈ C(V ) and for any x ∈ V , we have that L(u)(x) = P y∈V c(x, y)(u(x) − u(y)). Moreover, for u, v ∈ C(V ), the following holds: (i) First Green Identity: Z 1 Z vL(u)dx = h∇u, ∇vidx V 2 V

1 Z = c(x, y)u(x) − u(y)v(x) − v(y)dxdy. 2 V ×V R R (ii) Second Green Identity: V vL(u)dx = V uL(v)dx; R (iii) Gauss Theorem: V L(u)dx = 0. Proof. The expression for the Laplacian of u follows from the expression of the divergence, keeping in mind that ∇u is a ﬂow. On the other hand, given v ∈ C(V ), from the deﬁnition of divergence we get that Z Z 1 Z vL(u) dx = − vdiv (∇u) dx = h∇u, ∇vidx V V 2 V

1 Z = c(x, y)u(x) − u(y)v(x) − v(y)dx dy, 2 V ×V and the ﬁrst Green identity follows. The proof of the second Green identity and the Gauss theorem are then straightforward consequences. 5.3. Diﬀerence Operators on Networks 181

Corollary 5.3.5. The Laplacian of G is self-adjoint and positive semideﬁnite. More- over, L(u) = 0 if and only if u is constant.

Remark 5.3.6. Suppose that V = {x1, . . . , xn} and consider cij = c(xi, xj) = cji. T n Then, each u ∈ C(V ) is identified with u(x1), . . . , u(xn) ∈ R and the Laplacian of G is identified with the irreducible matrix   κ1 −c12 · · · −c1n    −c21 κ2 · · · −c2n    L =  . . . .  ,  . . .. .    −cn1 −cn2 ··· κn Pn where κi = j=1 cij, for i = 1, . . . , n. Clearly, this matrix is symmetric and diagonally dominant and, hence, it is positive semidefinite. Moreover, it is singular and 0 is a simple eigenvalue whose associated eigenvectors are constant.

5.3.1 Schr¨odingerOperators

A Schr¨odingeroperator on G is a linear operator Lq : C(V ) → C(V ) assigning to each u ∈ C(V ) the function Lq(u)(x) = L(u)(x) + q(x)u(x), where q ∈ C(V ) is called the potential. The bilinear form Eq(u, v) = hu, Lq(v)i is called the energy of Lq. Notice that, from the ﬁrst Green identity, Z Z 1 Eq(u, v) = c(x, y) u(x) − u(y) v(x) − v(y) dxdy + quv. 2 V ×V V

The fundamental problem in this framework is to determine when Eq is positive semidefinite; that is, when Eq(u) = Eq(u, u) ≥ 0 for any u ∈ C(V ). Observe that the matrix associated with Lq is   κ1 + q(x1) −c12 · · · −c1n    −c12 κ2 + q(x2) · · · −c2n    Lq =  . . . .  ,  . . .. .    −c1n −c2n ··· κn + q(xn) which is an irreducible and symmetric Z-matrix. Therefore, the fundamental problem is equivalent to determining when Lq is an M-matrix. Of course, this is the case when q ≥ 0, Lq is a non-singular M-matrix, and q is non-null. In order to obtain the necessary and sufficient condition for the positive semi- definiteness of Schrödingeroperators, it will be useful to consider the so-called Doob transform, which is a common tool in the framework of Dirichlet forms. We introduce the following concept: a (vertex) weight is a function σ ∈ C(V ) such that σ > 0 on V and hσ, σi = 1. The set of weights is denoted by Ω(V ). Given a weight 182 Chapter 5. Boundary Value Problems on Finite Networks

σ on the set of vertices, we deﬁne the potential associated with σ as the function −1 qσ = −σ L(σ). Therefore, for any x ∈ V ,

1 X 1 X qσ(x) = − c(x, y) σ(x) − σ(y) = −κ(x) + c(x, y)σ(y). σ(x) σ(x) y∈V y∈V

R Since V σqσ dx = 0, qσ takes positive and negative values, except when σ is constant, in which case qσ = 0 and the corresponding Schr¨odingeroperator coincides with the Laplacian. Moreover, we prove that any potential q is closely related to a potential of the form qσ.

Lemma 5.3.7. Given q ∈ C(V ), there exist unique σ ∈ Ω(V ) and λ ∈ R such that q = qσ + λ.

Proof. If we consider the matrix M = tI − Lq, for large enough t ∈ R, then M is an irreducible, symmetric, and nonnegative matrix. From the Perron–Frobenius theorem, the greatest eigenvalue of M, say µ, is simple and positive and has an associated eigenvector that is also positive, σ. Therefore, Lq(σ) = (t − µ)σ and it suﬃces to take λ = t − µ.

Suppose that there exist α ∈ R and ω ∈ Ω(V ), such that q = qσ + λ = qω +α. Therefore, Lq(σ) = λσ and Lq(σ) = ασ hence, applying the second Green’s R Identity, (λ − α) V σω dx = 0, which implies that λ = α. Again considering M = tI − Lq, we get that M(σ) = (t − λ)σ and M(ω) = (t − λ)ω, and hence ω = σ, since M is an irreducible, symmetric, and nonnegative matrix.

Proposition 5.3.8 (Doob Transform). Given σ ∈ Ω(V ) and for any u ∈ C(V ), the following identity holds:

1 Z u(x) u(y) L(u)(x) = c(x, y)σ(x)σ(y) − dy − qσ(x) u(x), σ(x) V σ(x) σ(y) for x ∈ V . In addition, for any u, v ∈ C(V ) we get that

1 Z Z u(x) u(y) v(x) v(y) Z E(u, v) = c(x, y)σ(x)σ(y) − − dxdy − qσu v. 2 V V σ(x) σ(y) σ(x) σ(y) V

Proof. First observe that

u(x) u(y) σ(x)u(x) − u(y) = σ(x)σ(y) − + σ(x) − σ(y) u(x), σ(x) σ(y) 5.3. Diﬀerence Operators on Networks 183 for any x, y ∈ V . So, if x ∈ V , then Z L(u)(x) = c(x, y)u(x) − u(y) dy V

1 Z u(x) u(y) = c(x, y)σ(x)σ(y) − dy σ(x) V σ(x) σ(y)

u(x) Z + c(x, y)σ(x) − σ(y)dy σ(x) V

1 Z u(x) u(y) = c(x, y)σ(x)σ(y) − dy − qσ(x) u(x). σ(x) V σ(x) σ(y)

Finally, we get that

Z v(x) Z u(x) u(y) Z E(u, v) = c(x, y)σ(x)σ(y) − dydx − qσu v V σ(x) V σ(x) σ(y) V

Z Z v(x) u(x) u(y) Z = c(x, y)σ(x)σ(y) − dxdy − qσu v. V V σ(x) σ(x) σ(y) V

Therefore, the last claim is a consequence of the following identities,

Z Z v(x) u(x) u(y) c(x, y)σ(x)σ(y) − dxdy V V σ(x) σ(x) σ(y)

Z Z v(y) u(y) u(x) = c(y, x)σ(y)σ(x) − dydx V V σ(y) σ(y) σ(x)

Z Z v(y) u(x) u(y) = − c(x, y)σ(x)σ(y) − dxdy, V V σ(y) σ(x) σ(y) where we have taken into account the symmetry of c.

Corollary 5.3.9 (The energy principle). If q = qσ + λ, σ ∈ Ω(V ), and λ ∈ R, then Lq is positive semideﬁnite if and only if λ ≥ 0, and positive deﬁnite if and only if λ > 0. Moreover, when λ = 0, Lq(u) = 0 if and only if u = aσ, a ∈ R.

Proof. Since q = qσ + λ, from the above proposition, we get that

Z Z 2 Z 1 u(x) u(y) 2 Eq(u) = c(x, y)σ(x)σ(y) − + λ u . 2 V V σ(x) σ(y) V 184 Chapter 5. Boundary Value Problems on Finite Networks

In particular, Eq(σ) = λ. Therefore, Lq is positive semideﬁnite if and only if λ ≥ 0 and positive deﬁnite if and only if λ > 0. When λ = 0, Lq(u) = 0 if and only if

1 Z Z u(x) u(y) 2 0 = Eq(u) = c(x, y)σ(x)σ(y) − ; 2 V V σ(x) σ(y) that is, if and only if u = aσ since G is a connected network.

Observe that minhu,ui=1{Eq(u)} ≥ λ, and Eq(u) = λ if and only if u = σ. Therefore, Lq(σ) = λσ, λ is the lowest eigenvalue of Lq and it is simple. The matrix version of the former formulation and results is that   d1 −c12 · · · −c1n    −c12 d2 · · · −c2n    M =  . . . .   . . .. .    −c1n −c2n ··· dn is an M-matrix if and only if there exist σ ∈ Ω(V ) and λ ≥ 0 such that di = 1 Pn λ + cijσj, where cij = cji when i > j. Moreover, M is invertible if and σi j=1, j6=i only if λ > 0. Equivalently, M is an M-matrix if and only if there exists σ ∈ Ω(V ) 1 Pn such that di ≥ cijσj. σi j=1, j6=i The concept of Schrödingeroperator encompasses other widely used discrete operators such as the normalized Laplacian introduced in Chung–Langlands [20], defined as ! 1 X u(x) u(y) ∆(u)(x) = p c(x, y) p − p . κ(x) y∈V κ(x) κ(y) As we can see, the normalized Laplacian is nothing else but a Schrödingeroperator associated with a new network. Specifically, if we denote by Lb the combinatorial p p Laplacian of the network Gb = (V,E, c) where c = c(x, y)/ κ(x) κ(y), then b √ b (5.3.8) implies that ∆ = Lbqσ , where σ = κ/2m and m denotes the size of G.

5.4 Glossary

Let M = (mij) be a symmetric square matrix of order n. Then (see Ben-Israel– Greville [10] for all the deﬁnitions given here), (i) (Diagonally dominant (d.d.) matrix) M is a diagonally dominant ma- Pn trix if and only if |mii| ≥ j=1, i6=j |mij| and it is strictly diagonally dominant Pn if |mii| > j=1, i6=j |mij|, for any i; (ii) (Generalized inverse) Mg is a generalized inverse of M if and only if MMgM = M; 5.5. Networks with Boundaries 185

(iii) (Group Inverse) M# is the group inverse of M if and only if MM#M = M, M#MM# = M#, and M#M = MM#; in general, the inverse group matrix does not exist for any squared matrix but, when M is a symmetric matrix, then M# exists and M# = M†. (iv) (Irreducible) M is reducible if there exists a permutation matrix P such h A 0 i that P>MP = , otherwise M is irreducible; 0 B (v) (Jacobi matrix) M is a Jacobi matrix if and only if it is a tridiagonal matrix; (vi) (M-matrix) M is an M-matrix if and only if it is a positive semideﬁnite Z-matrix; in particular, mii ≥ 0, for any (vii) (Moore–Penrose Inverse) M† is the Moore–Penrose inverse of M if and only if MM†M = M, M†MM† = M†,(M†M)> = M†M, and (MM†)> = MM†; moreover, M is symmetric if and only if M† is symmetric and then MM† = M†M;

(viii) (Positive) M is a non-negative matrix, denoted M ≥ 0, if and only if mij ≥ 0; M is a positive matrix, denoted M > 0, if and only if mij > 0; (ix) (Stieltjes matrix) M is a Stieltjes matrix if and only if it is a positive deﬁnite M-matrix; (x) (Strictly ultrametric matrix) M is a strictly ultrametric matrix if and only if M is nonnegative, mij ≥ min{mik, mjk} for any i, j, k, and mii > mij for all i =6 j;

(xi) (Z-matrix) M is a Z-matrix if and only if mij ≤ 0 for any i =6 j; i;

5.5 Networks with Boundaries

From now on we suppose the network (G, c) and the associated inner products on C(V ) and X (G), to be ﬁxed. Given a vertex subset F ⊂ V , we denote its complement in V by F c and its characteristic function by χF . For any x ∈ V , εx denotes the characteristic function ◦ of {x}. Moreover, we deﬁne the following sets: the interior of F , F = {x ∈ F | {y ∼ x} ⊂ F }, the (vertex) boundary of F , δ(F ) = {x ∈ F c | ∃y ∈ F such that y ∼ x}, and the closure of F , F¯ = F ∪ δ(F ). Figure 5.3 shows the above sets for a given network and F . If F ⊂ V is a proper subset, we say that F is connected if, for any x, y ∈ V , there exists a path joining x and y whose vertices are all in F . It is easy to prove that F¯ is connected when F is. In the following discussion, we always assume that F is a connected set. Moreover, if F ⊂ V , C(F ) denotes the subspace of C(V ) formed by the functions whose support is contained in F . We are also interested in the divergence theorem and the Green identities, which play a fundamental role in the analysis of boundary value problems. These 186 Chapter 5. Boundary Value Problems on Finite Networks

F d(F)

◦ Figure 5.3: Blue: F , Green: δ(F ), Circle: F

Figure 5.4: Normal vector field to F results are given on a finite vertex subset, the discrete equivalent to a compact manifold with a boundary, so we need to define the discrete analog of the outer normal vector field to the set.

The normal vector field to F is defined as nF = −dχF . Therefore, the compo- c nent function of nF is given by nF (x, y) = 1 when y ∼ x and (x, y) ∈ δ(F )×δ(F ), c nF (x, y) = −1 when y ∼ x and (x, y) ∈ δ(F ) × δ(F ), and nF (x, y) = 0 otherwise. c In consequence, nF c = −nF and supp(nF ) = δ(F ) ∪ δ(F ); see Fig. 5.4 Of course, networks do not have boundaries by themselves, but starting from ¯ a network we can define a network with a boundary as G = (F , cF ), where F is a connected proper subset and cF = c · χ(F¯×F¯)\(δ(F )×δ(F )) . From now on, we will work with networks with boundaries. Moreover, for the sake of simplicity, we use the notation c = cF . 5.5. Networks with Boundaries 187

Corollary 5.5.1 (Divergence Theorem). For any f ∈ X (G), we have that

Z Z a div f dx = (f , nF ) dx, F δ(F )

P where (f, g)(x)= y∈V f(x, y)g(x, y) denotes the standard inner product on Tx(G).

Proof. Taking u = χF in the deﬁnition of div we get

Z Z Z Z 1 1 a div (f) dx = χF div (f) dx = − hf, ∇χF i dx = − hf , ∇χF i dx F V 2 V 2 V Z Z Z 1 a 1 a 1 a = f , nF dx = f , nF dx + f , nF dx. 2 V 2 δ(F ) 2 δ(F c)

The result follows, taking into account that

Z a X X a f , nF dy = f (y, x)nF (y, x) c δ(F ) y∈δ(F c) x∈δ(F )

Z X X a a = f (x, y)nF (x, y) = f , nF dx. x∈δ(F ) y∈δ(F c) δ(F )

Recall that the Laplacian of G is the linear operator L: C(F¯) → C(F¯) as- ¯ P signing to each u ∈ C(F ) the function L(u)(x) = y∈F¯ c(x, y) u(x) − u(y) , for x ∈ F¯. Given q ∈ C(F¯), the Schr¨odingeroperator on G with potential q is the linear operator Lq : C(F¯) → C(F¯) assigning to each u ∈ C(F¯) the function Lq(u) = L(u) + qu. For each u ∈ C(F¯) we deﬁne the normal derivative of u on F as the function in C(δ(F )) given by

∂u X (x) = ∇u, n (x) = c(x, y) u(x) − u(y), ∂n F F y∈F

¯ for any x ∈ δ(F ). The normal derivative on F is the operator ∂/∂nF : C(F ) → C(δ(F )) assigning to any u ∈ C(F¯) its normal derivative on F . The relations between the values of the Schr¨odingeroperator with potential q on F and the values of the normal derivative at δ(F ) is given by the following identities. 188 Chapter 5. Boundary Value Problems on Finite Networks

Proposition 5.5.2. Given u, v ∈ C(F¯) the following properties hold: (i) First Green Identity: Z Z 1 vLvq(u)dx = c(x, y) u(x) − u(y) v(x) − v(y) dxdy F 2 F¯×F¯

Z Z ∂u + quvdx − v dx. F δ(F ) ∂nF

(ii) Second Green Identity: Z Z ∂v ∂u vLq(u) − uLq(v) dx = u − v dx. F δ(F ) ∂nF ∂nF

(iii) Gauss Theorem: R L(u)dx = − R ∂u dx. F δ(F ) ∂nF Proof. We get that Z Z Z vL(u) dx = c(x, y)v(x)u(x) − u(y)dydx F F F¯

Z Z = c(x, y)v(x)u(x) − u(y)dydx F¯ F¯

Z Z − c(x, y)v(x)u(x) − u(y)dydx δ(F ) F¯

Z Z ∂u = c(x, y)v(x)u(x) − u(y)dydx − v dx F¯×F¯ δ(F ) ∂nF

1 Z Z ∂u = c(x, y)u(x) − u(y)v(x) − v(y)dydx − v dx, 2 F¯×F¯ δ(F ) ∂nF and the ﬁrst Green identity follows. The proofs of the second Green identity and the Gauss theorem are straightforward consequences of the ﬁrst Green identity.

5.6 Self-Adjoint Boundary Value Problems

Given δ(F ) = H1 ∪H2 a partition of δ(F ), and functions q ∈ C(F ∪H1), g ∈ C(F ), g1 ∈ C(H1), and g2 ∈ C(H2), a boundary value problem on F consists of ﬁnding u ∈ C(F¯) such that ∂u Lq(u) = g on F, + qu = g1 on H1, and u = g2 on H2. (5.4) ∂nF 5.6. Self-Adjoint Boundary Value Problems 189

The associated homogeneous boundary value problem consists of finding u ∈ C(F¯) such that ∂u Lq(u) = 0 on F, + qu = 0 on H1, and u = 0 on H2. ∂nF It is clear that the set of solutions of the homogeneous boundary value problem is a vector subspace of C(F ∪ H1), which we denote by V. Moreover, if problem (5.4) has a solution u, then u + V describes the set of all its solutions. Problem (5.4) is generically known as a mixed Dirichlet–Robin problem, speci- ficly when H1,H2 =6 ∅ and q =6 0 on H1, and summarizes the different boundary value problems that appear in the literature with the following names:

(i) the Dirichlet problem: ∅= 6 H2 = δ(F ) and hence H1 = ∅;

(ii) the Robin problem: ∅= 6 H1 = δ(F ) and q =6 0 on H1;

(iii) the Neumann problem: ∅= 6 H1 = δ(F ) and q = 0 on H1;

(iv) the mixed Dirichlet–Neumann problem: H1,H2 =6 ∅ and q = 0 on H1;

(v) the Poisson equation on V : H1 = H2 = ∅ and hence F = V . Applying the Second Green Identity, we can show that the raised boundary value problem has some sort of symmetry. In addition, we obtain the conditions assuring the existence and uniqueness of solutions to the boundary value problem (5.4). Proposition 5.6.1. The boundary value problem (5.4) is self-adjoint; that is, for any R u, v ∈ C(F ∪H1) such that ∂u/∂n +qu = ∂v/∂n +qv = 0, we have vLq(u)dx = R F F F F uLq(v)dx. Proposition 5.6.2 (Fredholm Alternative). The boundary value problem (5.4) has a R R R solution if and only if gv dx+ g1v dx = g2∂v/∂n dx, for each v ∈ V. In F H1 H2 F addition, when the above condition holds, there exists a unique solution u ∈ C(F¯), R such that F¯ uv dx = 0, for any v ∈ V. Proof. First, observe that problem (5.4) is equivalent to the boundary value problem

∂u ∂g2 Lq(u) = g − L(g2) on F, + qu = g1 − on H1, and u = 0 on H2 ∂n ∂n F F in the sense that u is a solution of this problem if and only if u + g2 is a solution of (5.4). Now consider the linear operator F : C(F ∪ H1) → C(F ∪ H1) deﬁned as  L(u) + qu on F,  F(u) = ∂u  + qu on H1. ∂n F 190 Chapter 5. Boundary Value Problems on Finite Networks

Then, kerF = V and, moreover, by applying Proposition 5.6.1 for any u, v ∈ C(F ∪ H1) one can see that ! Z Z Z ∂u vF(u) dx = vLq(u) dx + v + qu dx ∂n F ∪H1 F δ(F ) F

! Z Z ∂v Z = uLq(v) dx + u + qv dx = uF(v) dx. ∂n F δ(F ) F F ∪H1

Therefore, the operator F is self-adjoint with respect to the inner product induced ⊥ in C(F ∪H1), and hence ImF = V by applying the classical Fredholm alternative. Consequently, Problem (5.4) has a solution if and only if the functiong ˜ ∈ C(F ∪ H ), given byg ˜ = g − L(g ) on F andg ˜ = g − ∂g /∂n on H , veriﬁes that 1 2 1 2 F 1 Z Z Z Z Z ∂g2 0 = gv˜ dx = gv dx + g1v dx − vL(g2) dx − v dx ∂n F ∪H1 F H1 F H1 F

Z Z Z Z ∂v = gv dx + g1v dx − g2L(v) dx − g2 dx ∂n F H1 F δ(F ) F

Z Z Z ∂v = gv dx + g1v dx − g2 dx, ∂n F H1 H2 F for any v ∈ V. Finally, when the necessary and suﬃcient condition is attained, there exists ⊥ a unique w ∈ V such that F(w) =g ˜. Therefore, u = w +g2 is the unique solution to Problem (5.4) such that, for any v ∈ V, Z Z Z uv dx = uv dx = wv dx = 0, F¯ F ∪H1 F ∪H1 since v = 0 on H2, and g2 = 0 on F ∪ H1. Observe that as a byproduct of the above proof, we obtain that uniqueness is equivalent to existence for any data. Next, we establish the variational formulation of the boundary value problem (5.4), representing the discrete version of the weak formulation for boundary value problems. Before describing the claimed formulation, we give some useful deﬁnitions. The bilinear form associated with the boundary value problem (5.4) is denoted by B : C(F¯) × C(F¯) → R, and it is given by

Z Z ∂u Z B(u, v) = vLq(u) dx + v dx + q uv dx; ∂n F δ(F ) F H1 5.6. Self-Adjoint Boundary Value Problems 191 hence, from the Second Green Identity, B(u, v) = B(v, u) for any u, v ∈ C(F¯), i.e., B is symmetric. In addition, by applying the First Green Identity, we obtain 1 Z Z Z B(u, v) = c(x, y) u(x)−u(y)v(x)−v(y) dxdy+ q uv dx+ q uv dx. 2 F¯×F¯ F H1

Associated with any pair of functions g ∈ C(F ) and g1 ∈ C(H1) we deﬁne the R R linear functional `: C(F¯) → R as `(v) = gv dx + g1v dx, whereas for any F H1 function g2 ∈ C(H2) we consider the convex set Kg2 = g2 + C(F ∪ H1).

Proposition 5.6.3 (Variational Formulation). Given g ∈ C(F ), g1 ∈ C(H1) and g2 ∈ C(H2), u ∈ Kg2 is a solution to Problem (5.4) if and only if B(u, v) = `(v) for any v ∈ C(F ∪H1) and, in this case, the set of all solutions to (5.4) is u+{w ∈ C(F ∪ H1) | B(w, v) = 0, ∀v ∈ C(F ∪ H1)}.

Proof. A function u ∈ Kg2 satisﬁes B(u, v) = `(v) for any v ∈ C(F ∪ H1) if and only if ! Z Z ∂u v(Lq(u) − g) dx + v + qu − g1 dx = 0. ∂n F H1 F ∗ Then, the ﬁrst result follows by taking v = εx, x ∈ F ∪ H1. Finally, u ∈ Kg2 is ∗ another solution to (5.4) if and only if B(u , v) = `(v) for any v ∈ C(F ∪ H1) and, ∗ hence, if and only if B(u − u , v) = 0 for any v ∈ C(F ∪ H1).

Observe that the equality B(u, v) = `(v) for any v ∈ C(F ∪ H1) assures that the condition of existence of solutions given by the Fredholm alternative holds since, for any v ∈ C(F ∪ H1), we have that Z Z gv dx + g1v dx = B(u, v) = B(v, u) F H1 Z Z ∂v Z = uLq(v) dx + u dx + q uv dx. ∂n F δ(F ) F H1 In particular, if v ∈ V we get that Z Z Z ∂v gv dx + g1v dx = g2 dx. ∂n F H1 H2 F

On the other hand, we note that the vector subspace {w ∈ C(F ∪ H1) | B(w, v) = 0, ∀v ∈ C(F ∪ H1)} is precisely the set of solutions to the homogeneous boundary value problem associated with (5.4). So, Problem (5.4) has a solution for any values of g, g1, and g2 if and only if it has a unique solution and this occurs if and only if w = 0 is the unique function in C(F ∪ H1) such that B(w, v) = 0, for any v ∈ C(F ∪ H1). Therefore, to assure the existence (and uniqueness) of solutions to Problem (5.4) for any data, it suﬃces to provide conditions under which B(w, w) = 0, with w ∈ C(F ∪ H1), implies that w = 0. In particular, this is the case when B is positive deﬁnite on C(F ∪ H1). 192 Chapter 5. Boundary Value Problems on Finite Networks

The quadratic form associated with the boundary value problem (5.4) is the function Q: C(F¯) → R given by Q(u) = B(u, u), i.e.,

1 Z 2 Z Z Q(u) = c(x, y) u(x) − u(y) dxdy + q u2dx + q u2dx. 2 F¯×F¯ F H1 Corollary 5.6.4 (Dirichlet Principle). Assume that Q is positive semi-deﬁnite on C(F ∪ H1). Let g ∈ C(F ), g1 ∈ C(H1), g2 ∈ C(H2) and consider the quadratic ¯ R functional J : C(F ) → given by J (u) = Q(u) − 2`(u). Then, u ∈ Kg2 is a solution to Problem if and only if it minimizes J on . (5.4) Kg2

Proof. First note that when u ∈ Kg2 , then Kg2 = u + C(F ∪ H1).

If u is a minimum of J on Kg2 , then, for any v ∈ C(F ∪ H1), the function 2 ϕv : R −→ R given by ϕv(t) = J (u + tv) = J (u) + t Q(v) + 2t[B(u, v) − `(v)] 0 attains a minimum value at t = 0 and hence 0 = ϕv(0) = B(u, v)−`(v). Therefore, from Proposition 5.6.3, u is a solution to Problem (5.4). Conversely, if u ∈ Kg2 is a solution to Problem (5.4), then B(u, v) = `(v) for any v ∈ C(F ∪ H1), and hence J (u + v) = J (u) + Q(v) + B(u, v) − `(v) = J (u) + Q(v) ≥ J (u), since Q is positive semideﬁnite; that is, u is a minimum of J on Kg2 .

Notice that if Q is not positive semidefinite on F ∪ H1, then J cannot attain any minimum, since if there exists v ∈ C(F ∪ H1) such that Q(v) < 0, then limt→+∞ J (u + tv) = −∞. The adaptation of the Doob transform to networks with boundaries allows us to establish an easy sufficient condition to ensure that B is positive semidefinite. Given a weight σ ∈ Ω(F¯), we define the potential associated with σ as the function q = −σ−1L(σ) on F , and q = −σ−1∂σ/∂n on δ(F ). σ σ F Proposition 5.6.5 (Doob Transform). Given σ ∈ Ω(F¯), for any u ∈ C(F¯) the following identities hold: 1 Z u(x) u(y) L(u)(x) = c(x, y)σ(x)σ(y) − dy − qσ(x) u(x), x ∈ F, σ(x) F¯ σ(x) σ(y) and ∂u 1 Z u(x) u(y) (x) = c(x, y)σ(x)σ(y) − dy − qσ(x) u(x), x ∈ δ(F ). ∂nF σ(x) F σ(x) σ(y) In addition, for any u ∈ C(F¯) we get that Z Z 2 Z Z 1 u(x) u(y) 2 2 Q(u) = c(x, y)σ(x)σ(y) − dxdy+ (q−qσ)u − qσu . 2 F¯ F¯ σ(x) σ(y) F ∪H1 H2

Corollary 5.6.6 (The Energy principle). If there exist σ ∈ Ω(F¯) such that q ≥ qσ on F ∪ H1, then the energy Q is positive semideﬁnite on C(F ∪ H1). Moreover, it is not strictly deﬁnite if and only if H2 = ∅ and q = qσ, in which case Q(v) = 0 if and only if v = aσ, a ∈ R. 5.7. Monotonicity and the Minimum Principle 193 5.7 Monotonicity and the Minimum Principle

In this section we are concerned with either the Poisson equation or the Dirichlet problem. The main results can be found in [1, 3, 4]. So, we consider G either a network as in Section 5.3, and hence G = (V, c), or a network with a boundary as in Section 5.5, hence G = (F¯ , c). If q ∈ C(V ) is a potential, recall that the Poisson equation consists of, given f ∈ C(V ), ﬁnding u ∈ C(V ) such that

Lq(u) = f on V, (5.5) whereas the Dirichlet problem consists of, given f ∈ C(F ) and g ∈ C(δ(F )), ﬁnding u ∈ C(F¯) such that

Lq(u) = f on F, and u = g on δ(F ). (5.6)

We can treat both problems (5.5) and (5.6) in a unified manner given that the Poisson equation corresponds to the limit case of the Dirichlet problem when F = V or, equivalently, when δ(F ) = ∅. Consistently with the energy principles in Sections 5.3 and 5.5, from now on we assume that the potential satisfies q ≥ qσ for some weight σ ∈ Ω(F¯), so that Lq is positive semidefinite on C(F¯). A function u ∈ C(F¯) is called q-harmonic (resp., q-superharmonic, q-subharmonic) on F if and only if Lq(u) = 0 (resp., Lq(u) ≥ 0, Lq(u) ≤ 0) on F . Moreover, u ∈ C(V ) is called strictly q-superharmonic (resp., strictly q-subharmonic) on F if and only if Lq(u) > 0 (resp., Lq(u) < 0) on F . Proposition 5.7.1 (Hopf’s minimum principle). Let u ∈ C(F¯) be q-superharmonic on F and suppose that there exists x∗ ∈ F such that u(x∗) ≤ 0 and u(x∗)/σ(x∗) = miny∈F {u(y)/σ(y)}. Then, u = aσ, a ≤ 0, on F¯, u is q-harmonic on F , and either u = 0 or q = qσ on F . Proof. As u(x∗) ≤ 0, Z ∗ ∗ ∗ u(x ) u(y) ∗ ∗ ∗ 0 ≤ Lq(u)(x ) = c(x , y)σ(y) ∗ − dy + (q(x ) − qσ(x ))u(x ) ≤ 0, F¯ σ(x ) σ(y) which implies that Z ∗ ∗ u(x ) u(y) ∗ ∗ ∗ 0 = c(x , y)σ(y) ∗ − dy = (q(x ) − qσ(x ))u(x ). F¯ σ(x ) σ(y) From the first identity, u(y)/σ(y) = u(x∗)/σ(x∗) for all y ∈ F¯ with y ∼ x∗. Hence, u = aσ, a ∈ R, since F¯ is connected. Moreover, a ≤ 0, since u(x∗) ≤ 0. On the other hand, 0 ≤ Lq(u) = aLq(σ) = a(q − qσ)σ ≤ 0 on F , which implies that Lq(u) = 0 on F , and either a = 0 or q = qσ on F . Proposition 5.7.2 (The Monotonicity Principle). If u ∈ C(F¯) is q-superharmonic on F , the following results hold: 194 Chapter 5. Boundary Value Problems on Finite Networks

(i) if δ(F ) =6 ∅ and u ≥ 0 on δ(F ), then either u > 0 on F , or u = 0 on F¯;

(ii) if δ(F ) = ∅ and q =6 qσ, then either u > 0 on V , or u = 0 on V ; and

(iii) if δ(F ) = ∅ and q = qσ, then u = aσ for a ∈ R and hence u is q-harmonic. ∗ ∗ ∗ Proof. Let x ∈ F such that u(x )/σ(x ) = miny∈F {u(y)/σ(y)}. If δ(F ) = ∅ and q = qσ, then Z ∗ ∗ ∗ u(x ) u(y) 0 ≤ Lq(u)(x ) = c(x , y)σ(y) ∗ − dy ≤ 0, F¯ σ(x ) σ(y) which implies u = aσ for a ∈ R, since F¯ is connected. Moreover, u is q-harmonic. If u(x∗) > 0, then u > 0 on F . Otherwise, suppose that u(x∗) ≤ 0. Then, from Hopf’s minimum principle u = aσ, a ≤ 0, on F¯, Lq(u) = 0 on F , and either u = 0 or q = qσ on F . When δ(F ) =6 ∅, necessarily u = 0 on δ(F ) since u ≥ 0 on δ(F ), and hence u = 0 on F¯.

The next result shows that strictly qσ-superharmonic functions cannot have local minima on F , a well-known fact for the continuous case.

Proposition 5.7.3. If u ∈ C(F¯) is strictly qσ-superharmonic on F , then for any x ∈ F there exists y ∈ F¯ such that c(x, y) > 0 and u(y)/σ(y) < u(x)/σ(x). Proof. Let x ∈ F and suppose that, for all y ∈ F¯ with c(x, y) > 0, u(y)/σ(y) ≥ u(x)/σ(x). Then Z u(x) u(y) 0 < Lqσ (u)(x) = c(x, y)σ(y) − dy ≤ 0, F¯ σ(x) σ(y) which is a contradiction. In the case of networks with boundaries, i.e., δ(F ) =6 ∅, the Monotonicity Principle for qσ-superharmonic functions is equivalent to the following minimum principle. Proposition 5.7.4 (Minimum Principle). Let G = (F¯ , c) be a network with boundary and u ∈ C(F¯), qσ-superharmonic on F . Then min {u(x)/σ(x)} ≤ min{u(x)/σ(x)}, x∈δ(F ) x∈F and the equality holds if and only if u coincides on F¯ with a multiple of σ.

Proof. Consider m = minx∈δ(F ){u(x)/σ(x)} and v = u − mσ. Then, v is qσ- superharmonic on F since Lqσ (σ) = 0 and, moreover, v ≥ 0 on δ(F ). Therefore, the claims follows from Proposition 5.7.2 (i). Now, we obtain a new proof of the existence and uniqueness of solutions for the Dirichlet and Poisson problems, which includes a property of the support of the solution. 5.8. Green and Poisson Kernels 195

Corollary 5.7.5. Let G be a network such that when F = V , then q =6 qσ. For each f ∈ C(F ) there exists a unique u ∈ C(F ) such that Lq(u) = f on F . In addition, if f ∈ C+(F ), then u ∈ C+(F ) and supp(f) ⊂ supp(u).

Proof. Consider the endomorphism F : C(F ) → C(F ) given by F(u) = Lq(u)|F . By Proposition 5.7.2, F is a monotone operator; that is, if F(u) ≥ 0, then u ≥ 0. Therefore, if F(u) = 0, then F(−u) = 0, and hence u, −u ≥ 0, which implies that u = 0. So, F is injective, which implies that it is an isomorphism and, moreover, + + u ∈ C (F ) when f = Lq(u)|F ∈ C (F ). Also, if u(x) = 0, then f(x) = Lq(u)(x) = R − F¯ c(x, y) u(y) dy ≤ 0, so f(x) = 0.

5.8 Green and Poisson Kernels

In this section we assume that the potential satisfies q ≥ qσ for some weight σ ∈ Ω(F¯) so that Lq is positive semidefinite, and then we build the kernels associated with the inverse operators corresponding either to a semihomogeneous Dirichlet problem or to a Poisson equation. In the same way as in the continuous case, such operators will be called Green operators. In addition, for any proper subset, we will also consider the kernel associated with the inverse operator of the boundary value problem in which the equation is homogeneous and the boundary values are prescribed. Such an integral operator will be called a Poisson operator. Here, we study the properties of the above-mentioned integral operators. First, we establish the basic notions about integral operators and their associated kernels. Then, we prove the existence and uniqueness of Green and Poisson operators for each proper subset F , we show some of their properties, and we build the associated Green or Poisson kernels. Next, under the hypothesis q =6 qσ, we make an analogous study of the Green operator for V . Moreover, we also consider the singular case, that is, when both F = V and q = qσ, and we construct the Green operators representing the group inverse of the matrix associated with the boundary value problem. Given S, T ⊂ V , define C(S × T ) = {f : V × V → R | f(x, y) = 0 if (x, y) ∈/ S × T }. In particular, any function K ∈ C(F × F ) is called a kernel on F . x If K is a kernel on F , for each x, y ∈ F we denote by K and Ky the functions y of C(F ) defined by Kx(y) = K (x) = K(x, y). The integral operator associated with K is the endomorphism K: C(F ) → C(F ) assigning to each f ∈ C(F ) the function R K(f)(x) = F K(x, y) f(y) dy for all x ∈ V . Conversely, given an endomorphism K: C(F ) → C(F ), the associated kernel is given by K(x, y) = K(εy)(x). Clearly, kernels and operators can be identified with matrices, after labeling the vertex set. In addition, a function u ∈ C(F ) can be identified with the kernel K(x, x) = u(x) and K(x, y) = 0 otherwise, and hence with a diagonal matrix, which will be denoted by Du. When K is a kernel on F¯, for each x ∈ δ(F ) and each y ∈ F¯, we denote by y ¯ (∂K/∂nx)(x, y) the value (∂K /∂nF )(x), whereas, for each x ∈ F and y ∈ δ(F ), we 196 Chapter 5. Boundary Value Problems on Finite Networks

¯ denote by (∂K/∂ny)(x, y) the value (∂Kx/∂nF )(y). Clearly, ∂K/∂nx ∈ C(δ(F )×F ) and ∂K/∂ny ∈ C(F¯ × δ(F )), and hence both are kernels on F¯.

2 2 Lemma 5.8.1. If K is a kernel on F¯, then it satisﬁes ∂ K/∂nx∂ny =∂ K/∂ny∂nx ∈ C(δ(F ) × δ(F )). Moreover, for any x, y ∈ δ(F ),

∂2K Z (x, y) = κ(x)κ(y)K(x, y) − κ(x) c(y, z)K(x, z)dz ∂nx∂ny F Z Z Z − κ(y) c(x, z)K(z, y)dz + c(x, u)c(y, z)K(u, z) du dz. F F F

2 In addition, ∂ K/∂nx∂ny is a symmetric kernel when K is. Now we are ready to introduce the concept of Green operators and kernels. First, we consider the case in which G is either a network with a boundary or when F = V , then q =6 qσ. Recall that in this situation, the endomorphism F deﬁned in the proof of Corollary 5.7.5 as F(u) = Lq(u)|F is an isomorphism. Its inverse is called the Green operator of G and denoted by L−1. Therefore, when G is a network with a boundary, for any f ∈ C(F ), u = L−1(f) is the unique solution of the Dirichlet problem Lq(u) = f on F and u = 0 on δ(F ), whereas when F = V , for any f ∈ C(V ), u = L−1(f) is the unique solution of the Poisson equation Lq(u) = f on V . Observe that the matrix associated with the operator F is Lq(F ; F ) and its inverse, L−1, is the matrix associated with L−1. When G = (F¯ , c) is a network with a boundary, we call the linear operator P : C(δ(F )) → C(F¯) assigning to each g ∈ C(δ(F )) the unique function P(g) ∈ C(F¯), such that Lq(P(g)) = 0 on F and P(g) = g on δ(F ), the Poisson operator of G. In particular, when q = qσ, then P(σ) = σ. In the following result we investigate formal properties of the Green and Poisson operators.

Proposition 5.8.2. If either G is a network with a boundary or q =6 qσ when F = V , then the Green and Poisson operators of G are formally self-adjoint in the sense R −1 R −1 R that F g L (f) dy = F f L (g) dy for all f, g ∈ C(F ), and δ(F ) g P(f) dy = R δ(F ) f P(g) dy for all f, g ∈ C(δ(F )). Proof. Given f, g ∈ C(F ), consider u = L−1(f) and v = L−1(g). Then u, v ∈ C(F ), Lq(u) = f, and Lq(v) = g on F . In addition, since both the Dirichlet problem and the Poisson equation are formally self-adjoint, we get that Z Z Z Z −1 −1 g L (f) dy = u Lq(v) dy = v Lq(u) dy = f L (g) dy. F F F F

On the other hand, P is self-adjoint since it coincides with the identity operator on C(δ(F )). 5.8. Green and Poisson Kernels 197

The Green and Poisson operators of G are integral operators on F and F¯, respectively, so the kernels associated with them will be called Green and Poisson kernels of G, and denoted by L−1 and P , respectively. It is clear that L−1 ∈ C(F × F ) and P ∈ C(F¯ × δ(F )). Moreover, if f ∈ C(F ) R −1 and g ∈ C(δ(F )), then the functions given by u(x) = F L (x, y) f(y) dy and R ¯ v(x) = δ(F ) P (x, y) g(y) dy, for x ∈ F , are the solutions of the semihomogeneous boundary value problems Lq(u) = f on F , u = 0 on δ(F ), and Lq(v) = 0 on F , v = g on δ(F ), respectively. In particular, for each g ∈ C(δ(F )), we have g(x) = R R δ(F ) P (x, y) g(y) dy for all x ∈ δ(F ). So, 1 = δ(F ) P (x, y) dy for all x ∈ δ(F ) and R ¯ σ(x) = δ(F ) P (x, y) σ(y) dy for all x ∈ F when q = qσ. Now, the relationship between an integral operator and its associated kernel enables us to characterize the Green and Poisson kernels as solutions of suitable boundary value problems.

Proposition 5.8.3. If either G is a network with a boundary or q =6 qσ when F = V , −1 −1 then, for all y ∈ F , the function Ly is characterized by Lq(Ly ) = εy on F and for all y ∈ δ(F ), the function Py is characterized by Lq(Py) = 0 on F , and Py = εy on −1 −1 δ(F ). Moreover, L is symmetric on F , P (x, y) = εy(x) − (∂L /∂ny)(x, y) for 2 −1 all x ∈ F¯ and y ∈ δ(F ), and (∂P/∂nx)(x, y) = εy(x)κ(x) − (∂ L /∂nx∂ny)(x, y) for all x, y ∈ δ(F ). Therefore, (∂P/∂nx) is symmetric on δ(F ). −1 −1 −1 Proof. For each y ∈ F , Ly = L (εy), and hence Lq(Ly ) = εy. In the same way if y ∈ δ(F ), then Py = P(εy) is the unique solution to the boundary value problem Lq(Py) = 0 on F and Py = εy on δ(F ). Moreover, from Proposition 5.8.2, we get that, for all x, y ∈ F , Z Z −1 −1 −1 −1 −1 −1 L (x, y) = Ly (x) = εx L (εy) dz = εy L (εx) dz = Lx (y) = L (y, x). F F

On the other hand, u = Py is the unique solution to the boundary value problem Lq(u) = 0 on F and u = εy on δ(F ). This problem is equivalent to the semihomogeneous one Lq(v) = −Lq(εy) on F with v ∈ C(F ) and, hence, u = −1 Py = εy − L (L(εy)| ), for all y ∈ δ(F ). Since for all x ∈ F , L(εy)(x) = R F V c(x, z) εy(x)−εy(z) dz = −c(x, y), we get that for all x ∈ F , and all y ∈ δ(F ), Z −1 −1 L (L(εy)|F )(x) = − L (x, z) c(z, y) dz F

Z ∂L−1 = c(y, z) L−1(x, y) − L−1(x, z) dz = (x, y). F ∂ny

As (∂εy/∂nx)(x, y) = εy(x)κ(x), the expression for the normal derivative of Py and hence its symmetry follows from Lemma 5.8.1.

Proposition 5.8.4. If either G is a network with a boundary or q =6 qσ when −1 F = V , then for all y ∈ F , Ly > 0 on F for any y ∈ F , and 0 < Py < σ/σ(y) 198 Chapter 5. Boundary Value Problems on Finite Networks on F for any y ∈ δ(F ). Moreover, for all y ∈ F with F \{y} being connected, −1 −1 Ly < Ly (y)σ/σ(y) on F \{y}. Proof. A direct consequence of the monotonicity principle from Proposition 5.7.2 −1 is that Ly > 0 on F for any y ∈ F and that Py > 0 on F for any y ∈ δ(F ). Moreover, taking u = σ/σ(y) − Py, we get that Lq(u) = (q − qσ)σ/σ(y) ≥ 0 on F , and u(y) = 0, u > 0 on δ(F ) \{y}, and hence, by the monotonicity principle, u > 0 on F . Finally, we consider y ∈ F with H = F \{y} being connected, and v = −1 −1 Ly (y)σ/σ(y) − Ly . Applying the monotonicity principle to the network with a −1 boundary (H ∪δ(H), c), where δ(H) = δ(F )∪{y}, we get that Lq(v) = Ly (y)(q− qσ)σ/σ(y) ≥ 0 on H, v(y) = 0, v > 0 on δ(H) \{y}, and hence v > 0 on H. The above proposition tells us that the inverse of an M-matrix is always positive, a well-known fact, see Berman–Plemmons [11]. Moreover, it attains its maximum at the diagonal. Next, we deﬁne the concepts of Green operator and Green kernel when F = V and q = qσ. For this, we will consider the vectorial subspace V = ker(Lq) and π the orthogonal projection on it. We already know that V is the subspace generated by ⊥ σ, and hence π(f) = hf, σiσ. Recall that, Lq is an isomorphism of V . Moreover, for each f ∈ C(V ), there exists u ∈ C(V ) such that Lq(u) = f − π(f) and then u + V is the set of all functions such that Lq(v) = f − π(f). # We use the term Green operator for G to denote the operator Lq of C(V ) # ⊥ assigning to each f ∈ C(V ) the unique function Lq (f) ∈ V satisfying that # # Lq(Lq (f)) = f − π(f). Its associated kernel will be denoted by Lq . In this case, # the matrix associated with Lq is nothing else but the group inverse of Lq, usually # denoted by Lq . # Proposition 5.8.5. If F = V and q = qσ, then the Green operator Lq is self-adjoint # R and positive semideﬁnite. Moreover, hLq (f), fi = 0 if and only if f = aσ, a ∈ . # # ⊥ Proof. Let f, g ∈ C(V ), u = Lq (f), and v = Lq (g). Then, u, v ∈ V , Lq(u) = f − π(f), Lq(v) = g − π(g) and hence Z Z Z Z # g Lq (f) dx = g − π(g) u dx = u Lq(v) dx = v Lq(u) dx V V V V Z Z # = f − π(f) v dx = f Lq (g) dx. V V

# R # R Moreover, hLq (f), fi = V f Lq (f) dx = V u Lq(u) dx ≥ 0, with equality if and only if u = 0, which implies that f = π(f) and so f = aσ. # # If Lq is the Green kernel for G, then Lq ∈ C(V × V ) is symmetric and R # moreover, if f ∈ C(V ), then the function given by u(x) = V Lq (x, y) f(y) dy for ⊥ all x ∈ V is the unique solution in V to the Poisson equation Lq(u) = f − π(f). 5.9. The Dirichlet-to-Robin Map 199

The relationship between an integral operator and its associated kernel enables us again to characterize the Green kernel for G in terms of solutions to suitable boundary value problems. # Proposition 5.8.6. For all y ∈ V , the function (Lq )y is characterized by equations # R # # Lq((Lq )y) = εy − σ(y) σ and V σ (Lq )y dx = 0. Moreover, (Lq )y(y) > 0 and # # (Lq )y < (Lq )y(y)σ/σ(y) on V \{y}, for any y ∈ V with V \{y} being connected. # # Proof. Observe that π(εy) = σ(y)σ for any y ∈ V . Moreover, (Lq )y = Lq (εy) and # # # # hence Lq((Lq )y) = εy − σ(y)σ. As (Lq )y(y) = hεy, (Lq )y(εy)i > 0, since Lq is positive semideﬁnite and εy is not a multiple of σ. On the other hand, if F = V \{y} # # and u = (Lq )y(y)σ/σ(y) − (Lq )y, we get u(y) = 0 and Lq(u) = σ(y)σ > 0 on F . Applying the monotonicity principle, we get u > 0 on F .

5.9 The Dirichlet-to-Robin Map

In this section we define the Dirichlet-to-Robin map on general networks and study its main properties. This map measures the difference of voltages between boundary vertices when electrical currents are applied to them and hence it is the fundamental tool for the problem of finding the conductivities in a network, which arises in applications such as geophysical exploration (see Brown–Uhlmann [17]) and medical imaging or electrical impedance tomography, which is a promising non-invasive method of clinical diagnosis; see Borcea–Druskin–Guevara–Vasquez– Mamonov [14]. The Dirichlet-to-Robin map is naturally associated with a Schrö- dinger operator, and generalizes the concept of the Dirichlet-to-Neumann map for the case of the combinatorial Laplacian. Throughout this section we suppose that G = (F¯ , c) is a network with a boundary and that q = qσ + λχδ(F ), so that q ≥ qσ and the energy principle; see Corollary 5.6.6. Recall that the energy Eq : C(F¯) × C(F¯) → R is given by Z Z Z 1 Eq(u, v) = cF (x, y) u(x) − u(y) v(x) − v(y) dx dy + q u v, 2 F¯ F¯ F¯ for u, v ∈ C(F¯). From the First Green Identity, we get

Z Z ∂u Eq(u, v) = vLq(u) + v + qu (5.7) F δ(F ) ∂nF for any u, v ∈ C(F¯). Under the above hypothesis, for any g ∈ C(δ(F )), the Dirichlet problem Lq(u) = 0 on F and u = g on δ(F ) has ug = Pq(g) as its unique solution. The map Λq : C(δ(F )) → C(δ(F )) assigning to any function g ∈ C(δ(F )) the function Λq(g) = ∂ug/∂nF + qg is called the Dirichlet-to-Robin map. The Poisson kernel is directly related to the Dirichlet-to-Robin map Λq, as is shown in the proposition below. 200 Chapter 5. Boundary Value Problems on Finite Networks

Proposition 5.9.1. The Dirichlet-to-Robin map, Λq, is a self-adjoint, positive semi- R deﬁnite operator whose associated quadratic form is δ(F ) gΛq(g) = Eq(ug, ug). Moreover, λ is the lowest eigenvalue of Λq, and its associated eigenfunctions are multiples of σ. In addition, the kernel N ∈ C(δ(F ) × δ(F )) of Λq is

∂P ∂2L−1 N = q + q = κ + q − q , ∂nx ∂nx∂ny which is symmetric, negative oﬀ-diagonal, and positive on the diagonal. Proof. From (5.7) we get that, for any f, g ∈ C(δ(F )), Z Z fΛq(g) = Eq(uf , ug) = Eq(ug, uf ) = gΛq(f) δ(F ) δ(F ) and hence Λq is self-adjoint and positive semideﬁnite. Moreover, for every x ∈ ¯ δ(F ), using that Pq(σχδ(F ) ) = σ on F , it is easily seen that ∂σ Λq(σχδ(F ) )(x) = (x) + qσ(x)σ(x) + λσ(x) = λσ(x), ∂nF −1 since qσ = −σ ∂σ/∂nF on δ(F ). On the other hand, using Proposition 5.6.5 and taking into account that ug = g on δ(F ), we get Z Z 2 Z 1 ug(x) ug(y) 2 Eq(ug, ug) = cF (x, y)σ(x)σ(y) − dxdy + λ g 2 F¯ F¯ σ(x) σ(y) δ(F ) Z ≥ λ g2. δ(F )

The equality holds if and only if ug = aσ; that is, if and only if g = aσ. Suppose that g is a nonzero eigenfunction corresponding to the eigenvalue α. Then, by the deﬁnition of eigenvalue and the ﬁrst part of the proposition, Z Z Z 2 F 2 α g = gΛq(g) = Eq (ug, ug) ≥ λ g , δ(F ) δ(F ) δ(F ) which implies α ≥ λ. The expression and the symmetry property for the kernel follow from Proposi- tion 5.8.3. Finally, for any x, y ∈ δ(F ) with x =6 y, notice that Pq(x, y) = εy(x) = 0. In this case, we get that

∂Pq N(x, y) = Λq(εy)(x) = (x, y) ∂nx X X = c(x, z) Pq(x, y) − Pq(z, y) = − c(x, z)Pq(z, y) < 0, z∈F z∈F 5.10. Characterization of Symmetric M-Matrices as Resistive Inverses 201

P since Pq(z, y) > 0. Moreover, as Λq(σ) = λσ on δ(F ), x∈δ(F ) Λq(εy)(x)σ(x) = λσ(y) for any y ∈ δ(F ) and hence

−1 X Λq(εy)(y) = λ − σ(y) Λq(εy)(x)σ(x) > 0, x∈δ(F ) x6=y using the fact that Λq(εy)(x) < 0 for any x, y ∈ δ(F ), x =6 y, as shown above. The kernel of the Dirichlet-to-Robin map is closely related to the Schur complement of Lq |F in Lq; see Fallat–Johnson [25] and Curtis–Ingerman–Morrow [22, Thm. 3.2] for the combinatorial Laplacian and the Dirichlet-to-Neumann map.

Notice that the Robin problem, Lq(u) = f on F , ∂u/∂nF + qu = g on δ(F ), has the following matrix expression " #" # " # D −C vδ g L = = , −C> M v f where D is the diagonal matrix whose diagonal entries are given by κ + q, C = c(x, y) x∈δ(F ), y∈F , M is the matrix associated with Lq |F , and vδ, v, f, and g are the vectors determined by u , u , f, and g, respectively. Then, M is invertible |δ(F ) |F −1 −1 and M = Lq (x, y) x,y∈F . Moreover, the Schur complement of M in L is −1 > L/M = D − CM C = N(x, y) x,y∈δ(F ), since we have the equality given for N in Proposition 5.9.1 and the following −1 > 2 −1 equality from Lemma 5.8.1: CM C = (∂ Lq /∂nx∂ny)(x, y)x,y∈δ(F ), where we −1 have taken into account that Lq is symmetric and zero on δ(F ) × F .

5.10 Characterization of Symmetric M-Matrices as Resistive Inverses

The proof of the main result in this section is based on a commonly used technique in the context of electrical networks and Markov chains that, in fact, is used in Dellacherie–Mart´ınez–San-Mart´ın[23, 24]. We remark that in the probabilistic context, the function q is usually called the potential (vector) of the operator Lq; see, for instance, [24]. Given Lq, a positive definite Schrödingeroperator on G, the method consists of embedding the given network in a suitable host network. The new network is constructed by adding a new vertex, representing an absorbing state, joined to each vertex in the original network by a new edge whose conductance is the diagonal excess after the use of the h-transform; i.e., the Doob transform. Then the effective resistances of the network are used to study properties of the Green function, due to their relationships, enabling us to finding properties of the inverse of a matrix. So, let us start by describing the main facts about effective resistances. 202 Chapter 5. Boundary Value Problems on Finite Networks

5.10.1 The Kirchhoff Index and Effective Resistances The Kirchhoff index was introduced in chemistry as a better alternative to other parameters used for discriminating among different molecules with similar shapes and structures; see Klein–Randić[30]. Since then, a productive line of research has developed, and the Kirchhoff index has been computed for some classes of graphs with symmetries; see for instance Bendito–Carmona–Encinas–Gesto [5] and the references therein. This index is defined as the sum of all effective resistances between any pair of vertices of the network and is also known as the total resistance; see Ghosh–Boyd–Saberi [27], which coincides with the Kemeny constant when considering the Markov chain associated with the network. We have introduced a generalization of the Kirchhoff index of a finite network, which defines the effective resistance between any pair of vertices with respect to a value λ ≥ 0 and a weight σ on the vertex set. It turns out that λ is the lowest eigenvalue of a suitable semidefinite positive Schrödingeroperator and σ is the associated eigenfunction. Then we prove that the effective resistance, with respect to λ and σ, defines a distance on the network, as in the standard case, and hence it can be used for the same purposes. We show that the generalized effective resistance has analogous properties to those in the classical case. In particular, we obtain the relationship between the Kirchhoff index with respect to λ and σ and the eigenvalues of the associated Schrödingeroperator as well as the relationship between the effective resistances with respect to λ and σ and the eigenvalues and eigenfunctions of the aforementioned operator. In the standard setting, the effective resistance between vertices x and y is defined through the solution of the Poisson equation L(u) = f for the dipole with poles at x and y; that is, f = εx −εy. The knowledge of the effective resistance can be used to deduce important properties of electrical networks; see, for instance, [27, 30, 35]. In the sequel we work only with semidefinite positive Schrödingeroperators. Therefore, we will consider as fixed a value λ ≥ 0, a weight σ ∈ Ω(V ), and their associated potential q = qσ + λ. We can generalize the concept of effective resistance in the following way. Given x, y ∈ V , the σ-dipole between x and y is the function fxy = (εx − εy)/σ. Clearly, for any x, y ∈ V , π(fxy) = 0 and hence the Poisson equation Lq(v) = fxy is solvable and any solution maximizes the functional u(x) u(y) Z Jx,y(u) = 2 − − uLq(u) dx; σ(x) σ(y) V see Corollary 5.6.4. Given x, y ∈ V , the effective resistance between x and y with respect to λ and σ, is the value Rλ,σ(x, y) = maxu∈C(V ){Jx,y(u)}. Moreover, the Kirchhoff index of G, with respect to λ and σ, is the value

1 X 2 2 K(λ, σ) = Rλ,σ(x, y)σ (x) σ (y). 2 x,y∈V 5.10. Characterization of Symmetric M-Matrices as Resistive Inverses 203

The kernel Rλ,σ : V × V → R is called the eﬀective resistance of the network G with respect to λ and σ.

Proposition 5.10.1. If u ∈ C(V ) is a solution of the Poisson equation Lq(u) = R fxy, then Rλ,σ(x, y) = V uLq(u) dx = u(x)/σ(x) − u(y)/σ(y). Therefore, Rλ,σ is symmetric and nonnegative, and moreover Rλ,σ(x, y) = 0 if and only if x = y.

Proof. Given u ∈ C(V ) we get that Jx,y(u) = Jy,x(−u), and hence Rλ,σ(x, y) = Rλ,σ(y, x) for any x, y ∈ V . Moreover, we know that Rλ,σ(x, x) = 0 for any x ∈ V and also that Rλ,σ(x, y) = 0 if and only if hLq(u), ui = 0 for any solution of the Poisson equation Lq(u) = fxy. So, u = aσ, where a = 0 if λ > 0, which in any case implies that Lq(u) = 0 and hence fxy = 0 or, equivalently, x = y. The eﬀective resistance is closely related to the Green function of G. Recall −1 # that, when λ > 0, Lq is invertible and hence Lq = Lq . So, from now on, and for # the sake of simplicity, we consider only the kernel associated with Lq denoted by L# (notice that the dependence on q has been erased). Corollary 5.10.2. For any x, y ∈ V ,

L#(x, x) L#(y, y) 2L#(x, y) Rλ,σ(x, y) = + − . σ2(x) σ2(y) σ(x)σ(y)

In particular, K(λ, σ) = tr(L#)−λ†, where λ† = λ−1 if λ > 0 and λ† = 0 otherwise. # Proof. If u = L (fxy), then Lq(u) = fxy. Therefore, for any z ∈ V ,

Z # # # L (z, x) L (z, y) u(z) = L (z, t)fxy(t) dt = − . V σ(x) σ(y)

The result follows from the identity Rλ,σ(x, y) = u(x)/σ(x) − u(y)/σ(y), taking into account the symmetry of L#. On the other hand,

1 X L#(x, x) L#(y, y) 2L#(x, y) K(λ, σ) = + − σ(x)2σ(y)2 2 σ2(x) σ2(y) σ(x)σ(y) x,y∈V

1 X 1 X X = L#(x, x)σ2(y) + L#(y, y)σ2(x) − L#(x, y)σ(x)σ(y) 2 2 x,y∈V x,y∈V x,y∈V

X = L#(x, x) − hL#(σ), σi = tr(L#) − λ†. x∈V

The first result in the previous corollary shows that, for a constant weight σ and λ = 0, Rλ,σ(x, y) coincides with the definition in Section 4.8, page 154, up to a multiplicative factor n. Therefore, the value of K(λ, σ) is nK(G), as it is defined on page 156. 204 Chapter 5. Boundary Value Problems on Finite Networks

xˆ λω(x) x

Figure 5.5: Host network

5.10.2 Characterization

Given λ > 0, σ ∈ Ω(V ), andx ˆ ∈ / V , we consider the network Gλ,σ = (V ∪{xˆ}, cλ,σ), where cλ,σ(x, y) = c(x, y) when x, y ∈ V and cλ,σ(ˆx, x) = cλ,σ(x, xˆ) = λ σ(x) for λ,σ any x ∈ V . We denote by L its√ combinatorial Laplacian and by σb√∈ Ω(V ∪ {xˆ}) the weight given by σb(x) = 2σ(x)/2 when x ∈ V , and σb(ˆx) = 2/2; see Figure 5.5. The next result establishes the relationship between the original Schrödinger operator Lq and a new semidefinite Schrödingeroperator on Gλ,σ. λ,σ Proposition 5.10.3. If q = qσ + λ and we define qˆ = −L (σb)/σb, then qˆ(ˆx) = λ 1 − hσ, 1i and qˆ = q − λ σ on V . Moreover, for any u ∈ C(V ∪ {xˆ}) we λ,σ λ,σ get that Lqˆ (u)(ˆx) = λ(u(ˆx) − hσ, u|V i) and Lqˆ (u) = Lq(u|V ) − λ σ u(ˆx) = λ,σ Lq(u|V ) − λ Pσ(u|V ) − σ Lqˆ (u)(ˆx) on V . Proof. Given u ∈ C(V ∪ {xˆ}), then, for any x ∈ V , we get that

λ,σ L (u)(x) = L(u|V )(x) + λ σ(x)u(x) − λ σ(x)u(ˆx). In particular, taking u =σ ˆ, we obtain that −qˆ = −q + λ σ on V and hence λ,σ λ,σ Lqˆ (u)=Lq(u|V )−λ σ u(ˆx) on V . On the other hand, L (u)(ˆx)=λ (u(ˆx)h σ, 1i− λ,σ hσ, u|V i , which, in particular, implies −qˆ(ˆx) = Lqˆ (ˆσ)(ˆx) = λ(hσ, 1i − 1). There- λ,σ fore, for any u ∈ C(V ∪ {xˆ}), we get that Lqˆ (u)(ˆx) = λ (u(ˆx) − hσ, u|V i), which λ,σ is equivalent to λ σ u(ˆx) = σ Lqˆ (u)(ˆx) + λ Pσ(u|V ), and the second identity for λ,σ the value of Lqˆ (u) on V follows. We end this section with the matrix counterpart of the main results in Bendito–Carmona–Encinas–Gesto [6]. We characterize the inverse of any irreducible symmetric M-matrix, singular or not, in terms of the eﬀective resistances of a suitable network or, equivalently, we prove that any irreducible symmetric M- matrix is a resistive inverse. The interested reader can consult [6] for the proofs not given here. 5.10. Characterization of Symmetric M-Matrices as Resistive Inverses 205

Theorem 5.10.4. Let M be an irreducible Stieltjes matrix of order n and M−1 = (gij) be its inverse. Then, there exist a network G = (V, c) with |V | = n, a value

λ > 0, and a weight σ ∈ Ω(V ) such that M = Lqσ + λI. Moreover, if we consider the host network Gλ,σ = (V ∪ {xn+1}, bcλ,σ) and Rbij, i, j = 1, . . . , n + 1, are the eﬀective resistances of Gλ,σ with respect to σb, then " # " # M −Mb M −λb Lq = = , bσb −b∗M b∗Mb −λb∗ nλ

n where b ∈ R is the vector indentifed with σ and, for any i, j = 1, . . . , n, gij = σiσj(Rbin+1 + Rbjn+1 − Rbij)/2. The above theorem generalizes the main result obtained in Fiedler [26], where the inverses of weakly diagonal dominant Stieltjes matrices were characterized. Theorem 5.10.5. Let M be a singular irreducible and symmetric M-matrix of order # n and consider its group inverse M = (gij). Then, there exist a network G =

(V, c) with |V | = n and a weight σ ∈ Ω(V ) such that M = Lqσ . Moreover, if Rij, i, j = 1, . . . , n, are the eﬀective resistances of G with respect to σ, then n n σiσj 1 X 2 1 X 2 2 gij = − Rij − (Rik + Rjk)σ + Rklσ σ . 2 n k n2 k l k=1 k,l=1 Finally, the problem of knowing when the group inverse of a singular irreducible and symmetric M-matrix is also an M-matrix is diﬃcult. We have the following general result (see Bendito–Carmona–Encinas–Mitjana [8]), which is a consequence of the minimum principle and generalizes the results obtained in Kirkland–Neumann [28] for trees. Proposition 5.10.6. Let M be a singular irreducible and symmetric M-matrix. # Then, M = (gij) is an M-matrix if and only if gij ≤ 0 for any j ∼ i. Proof. If M is a singular irreducible and symmetric M-matrix, there exist a net- # # work G = (V, c) and a weight σ ∈ Ω(V ) such that M = Lqσ and M = L . # For any y ∈ V , consider uy(x) = −L (x, y). Then, uy is qσ-superharmonic on F = V \ ({y} ∪ N(y)), and hence minz∼y{uy(z)/σ(z)} ≤ minz∈F {uy(z)/σ(z)}, since δ(F ) = N(y); here, N(y) = {z ∈ V | z ∼ y} is the set of vertices adjacent to # # y. Therefore, maxz∼y{L (z, y)/σ(z)} ≥ L (x, y)/σ(x) for any x =6 y. We focus now on singular irreducible and symmetric Jacobi M-matrices. Given c1, . . . , cn−1 > 0 and d1, . . . , dn ≥ 0 such that the tridiagonal matrix   d1 −c1  −c1 d2 −c2     . . .  M =  ......  (5.8)    −cn−2 dn−1 −cn−1  −cn−1 dn 206 Chapter 5. Boundary Value Problems on Finite Networks is a singular M-matrix, we aim here to determine when its group inverse M# is also an M-matrix. We have proved that the matrix given in (5.8) is a singular M-matrix if and 2 2 only if there exist σ1, . . . , σn > 0 such that σ1 + ··· + σn = 1 and

c1σ2 cn−1σn−1 1 d1 = , dn = , and dj = (cjσj+1 + cj−1σj−1), (5.9) σ1 σn σj for any j =2, . . . , n−1. Moreover, the weight is uniquely determined by (d1, . . . , dn) and (c1, . . . , cn−1). n−1 In the sequel, the matrix given in (5.8), where c = (c1, . . . , cn−1)∈(0, +∞) is a conductance, σ ∈ Ω(V ), and the diagonal entries are given by (5.9), is denoted by M(c, σ) and its group inverse is denoted by M#(c, σ). Throughout this section, Pj Qj we use the conventions l=i al = 0 and l=i al = 1 when j < i. n In addition, ej denotes the j-th vector in the standard basis of R and e denotes the vector e = e1 + ··· + en. # Proposition 5.10.7. The group inverse of M(c, σ) is M (c, σ) = (gij), where

 k 2 n 2 n  P 2 P 2 P 2 i−1 σl n−1 σl j−1 σl X l=1 X l=k+1 X l=k+1  gji = gij = σiσj  + −   ckσkσk+1 ckσkσk+1 ckσkσk+1  k=1 k=i k=i  for any 1 ≤ i ≤ j ≤ n. Pn−1 Pn 2 Pk 2 Notice that g1n = − k=1 ( l=k+1 σl )( l=1 σl )/ckσkσk+1 < 0 and hence the group inverse of any path always has a negative entry. The group inverse for the normalized Laplacian, that is, when σ is the square root of the generalized degree, was obtained in Chung–Yau [21, Thm. 9]. If we take into account that the group inverse of a symmetric and positive semideﬁnite matrix is itself symmetric and positive semideﬁnite, as a byproduct of the expression of M#(c, σ) we can easily characterize when it is an M-matrix. # Theorem 5.10.8. M (c, σ) is an M-matrix if and only if gii+1 ≤ 0 for any i = 1, . . . , n − 1, that is, if and only if

n i k 2 n 2 P 2 P 2 P 2 P 2 σl σl i−1 σl n−1 σl l=i+1 l=1 X l=1 X l=k+1 ≥ + , i = 1, . . . , n − 1. ciσiσi+1 ckσkσk+1 ckσkσk+1 k=1 k=i+1

The above theorem for σ constant was given in Chen–Kirkland–Neumann [19, Lem. 3.1], where it is proved that n ≤ 4. For general weights, the above characterization involves a highly nonlinear system of inequalities on the oﬀ-diagonal entries of the matrix. The problem can be solved explicitly for n ≤ 3, but for n ≥ 4 the system becomes much more complicated and the key idea for solving it 5.11. Distance-regular Graphs with the M-Property 207 is to apply well-known properties of general M-matrices to the coeﬃcient matrix of the system. Our main result establishes that, for any n, there exist singular, symmetric and tridiagonal M-matrices of order n whose group inverse is also an M-matrix; see Bendito–Carmona–Encinas–Mitjana [9] for a complete study.

5.11 Distance-Regular Graphs with the M-Property

We aim here at characterizing when the group inverse of the combinatorial Lapla- cian matrix of a distance-regular graph is an M-matrix. In these cases, we say that the graph has the M-property. A connected graph G is called distance-regular if there are integers bi, ci, i = 0,...,D such that for any two vertices x, y ∈ V at distance i = d(x, y), there are exactly ci neighbors of y in Γi−1(x) and bi neighbors of y in Γi+1(x), where for any vertex x ∈ G the set of vertices at distance i from it is denoted by Γi(x). Moreover, |Γi(x)| will be denoted by ki. In particular, G is regular of degree k = b0. The sequence ι(Γ) = {b0, b1, . . . , bD−1; c1, . . . , cD} is called the intersection array of G. In addition, ai = k − ci − bi is the number of neighbors of y in Γi(x), for d(x, y) = i. Clearly, bD = c0 = 0, c1 = 1, and the diameter of G is D. Usually, the parameters a1 and c2 are denoted by λ and µ, respectively. For all the properties related to distance-regular graphs we refer the reader to [13, 16]. The parameters of a distance-regular graph have many relationships, among which we will make extensive use of the following:

(i) k0 = 1 and ki = b0 ··· bi−1/c1 ··· ci, for i = 1,...,D;

(ii) n = 1 + k + k2 + ··· + kD;

(iii) k > b1 ≥ · · · ≥ bD−1 ≥ 1;

(iv) 1 ≤ c2 ≤ · · · ≤ cD ≤ k;

(v) if i + j ≤ D, then ci ≤ bj and ki ≤ kj when, in addition, i ≤ j. Additional relationships between the parameters give more information on the structure of distance-regular graphs. For instance, if k = 2, then bi = 1, which implies that ci = 1 and ai = 0 for i = 1,...,D − 1. These graphs are called cycles. More precisely, if n ≥ 3, the n-cycle, Cn, is the distance-regular graph with n diameter D = 2 whose intersection array is ι(Cn) = {2, 1,..., 1; 1,..., 1, cD}, where cD = 1 when n is odd, and cD = 2 when n is even; see Brouwer–Cohen– Neumaier [16]. On the other hand, G is bipartite if and only if ai = 0, i = 1,...,D, whereas G is antipodal if and only if bi = cD−i, i = 0,...,D, i =6 bD/2c and then bbD/2c = kDcdD/2e and G is an antipodal (kD + 1)-cover of its folded graph; see [16, Prop. 4.2.2]. That is, the size of all maximal cliques of ΓD is kD + 1. Observe that Cn is antipodal if and only if n is even. It is well known that distance-regular graphs with degree k ≥ 3 other than bipartite and antipodal graphs are primitive, see for instance [16, Thm. 4.2.1]. In 208 Chapter 5. Boundary Value Problems on Finite Networks addition, any primitive distance-regular graph, in fact any non-antipodal distance- regular graph, satisﬁes k ≤ kD(kD − 1); see [16, Thm. 5.6.1]. The following lemma shows the group inverse for the combinatorial Laplacian of a distance-regular graph in terms of the parameters of its intersection array; see Bendito–Carmona–Encinas [2, Prop. 4.2] for the details. Lemma 5.11.1. Let G be a distance-regular graph. Then, for any x, y ∈ V ,

D−1 D D−1 r D # 1 X 1 X 1 X 1 X X L xy = kj − 2 ki ki . n krbr n krbr r=d(x,y) j=r+1 r=0 i=0 i=r+1

Proposition 5.11.2. A distance-regular graph G has the M-property if and only if

D−1 D X 1 X 2 n − 1 ki ≤ . k b k j=1 j j i=j+1

# Moreover, the subjacent graph of G is Kn when the above inequality is strict and G otherwise. Proof. From Proposition 5.10.6 and Lemma 5.11.1, the group inverse of L is an M-matrix if and only if

D−1 D 2 X 1 X n(n − 1) ki ≤ , k b k j=0 j j i=j+1 that is, if and only if

D−1 D 2 (n − 1)2 X 1 X n(n − 1) + ki ≤ . k k b k j=1 j j i=j+1

Finally, the above inequality is an equality if and only if G(x, y) = 0 when d(x, y) = 1, and hence the subjacent graph of G# is G. A distance-regular graph of order n has diameter D = 1 if and only if it is the complete graph Kn. In this case, the above inequality holds since the left side term vanishes. Therefore, any complete graph has the M-property. In fact, # 1 # L = n2 L (see Bendito–Carmona–Encinas–Mitjana [7]), and hence G is also a complete network. Corollary 5.11.3. If G has the M-property and D ≥ 2, then λ ≤ 3k−k2/(n−1)−n and hence n < 3k. Proof. When D ≥ 2, from the inequality in Proposition 5.11.2 we get that

D−1 D 2 (n − k − 1)2 X 1 X n − 1 ≤ ki ≤ . kb k b k 1 j=1 j j i=j+1 5.11. Distance-regular Graphs with the M-Property 209

2 Therefore, (n − 1 − k) ≤ (n − 1)b1 = (n − 1)(k − 1 − λ) and the upper bound for λ follows. In addition, this inequality implies that 0 ≤ λ < 3k − n and then 3k > n.

The inequality 3k > n turns out to be a strong restriction for a distance- regular graph to have the M-property. For instance, given n ≥ 3, if the n-cycle Cn has the M-property, necessarily 6 > n, which is true if and only if either D = 1 (i.e., n = 3) or D = 2 (i.e., n = 4, 5). Moreover, for n = 4, 5, Cn has the M- # 2 property since (L )ij = (n − 1 − 6|i − j|(n − |i − j|))/12n, i, j = 1, . . . , n; see, for instance, [7, 19]. In the following result, we generalize the above observation by showing that only distance-regular graphs with small diameters can satisfy the M-property.

Proposition 5.11.4. If G is a distance-regular graph with the M-property, then D ≤ 3.

Proof. If D ≥ 4, then, from property (v) of the parameters, k = k1 ≤ ki, i = 2, 3, and hence 3k < 1 + 3k ≤ 1 + k + k2 + k3 ≤ n; thus, G does not have the M- property.

5.11.1 Strongly Regular Graphs

A distance-regular graph whose diameter equals 2 is called a strongly regular graph. This kind of distance-regular graph is usually represented through the four parameters (n, k, λ, µ) instead of its intersection array; see Brouwer–Cohen–Neumaier [16]. Clearly, the four parameters of a strongly regular graph are not independent, since (n − 1 − k)µ = k(k − 1 − λ). (5.10)

For this reason some authors drop the parameter n in the above array; see, for instance, Biggs [13]. Moreover, equality (5.10) implies that 2k−n ≤ λ < k−1, since µ ≤ k and D = 2. Moreover, antipodal strongly regular graphs are characterized by satisfying µ = k or, equivalently, λ = 2k − n which, in particular, implies that 2k ≥ n. In addition, any bipartite strongly regular graph is antipodal and is characterized by satisfying µ = k and n = 2k. Observe that the only n-cycles satisfying the M-property are precisely C3, that is the complete graph with 3 vertices, and C4 and C5, which are strongly regular. In the following result, we characterize strongly regular graphs that have the M-property in terms of their parameters.

Proposition 5.11.5. A strongly regular graph with parameters (n, k, λ, µ) has the M-property if and only if µ ≥ k−k2/(n−1). In particular, every antipodal strongly regular graph has the M-property. 210 Chapter 5. Boundary Value Problems on Finite Networks

Proof. Clearly, for D = 2 the inequality in Corollary 5.11.3 characterizes the strongly regular graphs satisfying the M-property. The result follows, taking into account that, from equality (5.10),

k2 λ ≤ 3k − − n ⇐⇒ k(n − 1 − k) ≤ µ(n − 1). n − 1 Kirkland–Neumann [29, Thm 2.4] presents another characterization of strongly regular graphs with the M-property, in terms of the combinatorial Laplacian eigenvalues. It is straightforward to verify that the Petersen graph, whose parameters are (10, 3, 0, 1), does not have the M-property. So, it is natural to ask how many strongly regular graphs satisfy the above inequality. Before answering this question, we recall that if G is a primitive strongly regular graph with parameters (n, k, λ, µ), then its complement graph is also a primitive strongly regular graph, with parameters (n, n − k − 1, n − 2 − 2k + µ, n − 2k + λ) (see, for instance, Bondy– Murty [15]), which implies µ ≥ 2(k + 1) − n. Note that the complement of an antipodal strongly regular graph G is the disjoint union of m copies of a complete graph Kr for some positive integers m and r. Therefore, G is the complete multipartite graph Km×r. On the other hand, strongly regular graphs with the same parameters as their complements are called conference graphs, and their parameters are (4m + 1, 2m, m − 1, m), where m ≥ 1. Moreover, it is known that such a graph exists if and only if 4m + 1 is the sum of two squares; see Cameron [18]. Now we are ready to answer the question. Corollary 5.11.6. If G is a primitive strongly regular graph, then either G or G has the M-property. Moreover, both of them have the M-property if and only if G is a conference graph. Proof. If we deﬁne k¯ = n − k − 1, λ¯ = n − 2 − 2k + µ, andµ ¯ = n − 2k + λ, then k¯ − k¯2/(n − 1) = k − k2/(n − 1) and hence

k¯2 k2 k2 µ¯ ≥ k¯ − ⇐⇒ λ ≥ 3k − − n ⇐⇒ µ ≤ k − , n − 1 n − 1 n − 1 where the equality in the left side holds if and only if the equality in the right side holds. Moreover, any of the above inequalities is an equality if and only ifµ ¯ = µ and λ¯ = λ, that is, if and only if G is a conference graph. The remaining claims follow from Proposition 5.11.5.

5.11.2 Distance-regular Graphs with Diameter 3 In this section we characterize those distance-regular graphs with diameter D = 3 that also have the M-property. Distance-regular graphs with diameter 3 have been extensively treated in Biggs [12] and we refer the reader to this work, and references therein, for the main properties of such graphs. In this case, the intersection array 5.11. Distance-regular Graphs with the M-Property 211

is ι(Γ) = {k, b1, b2; 1, c2, c3}. Again, the parameters are not independent, since c2 divides kb1, c2c3 divides kb1b2 and, moreover, (n − 1 − k)c2c3 = kb1(b2 + c3). The next result follows straightforwardly from Proposition 5.11.2. Proposition 5.11.7. A distance-regular graph with diameter 3 has the M-property 2 2 2 2 if and only if k b1(b2c2 + (b2 + c3) ) ≤ c2c3(n − 1). To study distance-regular graphs with diameter 3 satisfying the M-property we recall that, according to Brouwer–Cohen–Neumaier [16, Thm. 4.2.1], these graphs are classified, in not necessarily disjoint classes, either as C6, C7, bipartite, antipodal, or primitive. Since neither C6 nor C7 satisfy the M-property we study the remaining classes starting with the bipartite and antipodal cases. The intersection array of a bipartite distance-regular graph with D = 3 is ι(Γ) = {k, k − 1, k − µ; 1, µ, k}, where 1 ≤ µ ≤ k − 1 and µ divides k(k − 1). These graphs are antipodal if and only if µ = k − 1. Otherwise, they are the incidence graphs of nontrivial square 2 − (n/2, k, µ) designs. Therefore, k − µ must be a square, see [16, Thm. 1.10.4]. Proposition 5.11.8. A bipartite distance-regular graph with D = 3 satisfies the M-property if and only if 4k/5 ≤ µ ≤ k − 1, and these inequalities imply k ≥ 5. Proof. From Corollary 5.11.3, G satisfies the M-property if and only if k(k − 1)(4k−5µ) ≤ µ2; this inequality holds if and only if 4k ≤ 5µ, otherwise 4k−5µ ≥ 1 and hence k(k − 1)(4k − 5µ) ≥ k(k − 1) > (k − 1)2 ≥ µ2. Finally, the inequality 4k/5 ≤ k − 1 implies k ≥ 5.

The above inequalities hold when µ = k −1 for k ≥ 5. Moreover, if 4k/5 = µ, then k = 5(4m + 1) and µ = 4(4m + 1), m ≥ 1, and k − µ is a square if and only if m = h(h + 1), h ≥ 1. Therefore, there exist an infinite family of arrays attaining the lower bound. On the other hand, choosing k = r2m+1 and µ = (r − 1)r2m with m ≥ 1 and r ≥ 6, we get another family that satisfies the condition with 4k/5 < µ < k − 1. If G is a bipartite distance-regular graph with D = 3 and µ < k − 1, it is well known (see, for instance, Brouwer–Cohen–Neumaier [16, p.17]) that Γ3 is also a bipartite distance-regular graph with D = 3 and whose intersection array is ι(Γ3) = {k3, k3 − 1, k − µ; 1, µ,¯ k3}, where k3 = (k − 1)(k − µ)/µ andµ ¯ = (k−µ−1)(k−µ)/µ. Then, Γ3 has the M-property if and only if 1 ≤ µ ≤ (k−1)/5. 2 Therefore, Γ3 has the M-property when k = m + 1 and µ = 1, m ≥ 2, or when k = 5m(m + 1) + 1 and µ = m(m + 1), m ≥ 1. On the other hand, given `, m ≥ 1 and r ≥ 2, if we consider k = r`(r`+2m +1) and µ = r`, giving another family that satisifies the condition for Γ3 having the M-property, with 1 < µ < (k − 1)/5. Corollary 5.11.9. If G is the bipartite distance-regular graph with intersection array ι(Γ) = {k, k − 1, k − µ; 1, µ, k}, where 1 ≤ µ < k − 1, then either G or Γ3 has the M-property, except when k − 1 < 5µ < 4k, in which case none of them has the M-property. 212 Chapter 5. Boundary Value Problems on Finite Networks

The intersection array of an antipodal distance-regular graph with D = 3 is ι(Γ) = {k, tµ, 1; 1, µ, k}, where µ, t ≥ 1 and tµ < k. These graphs have order n = (t + 1)(k + 1) and they are the (t + 1)-cover of the complete graph Kk+1. When t = 1, these antipodal distance-regular graphs are known as Taylor graphs, T (k, µ). Proposition 5.11.10. An antipodal distance-regular graph with D = 3 has the M- property if and only if it is a Taylor graph T (k, µ) with k ≥ 5 and (k+3)/2 ≤ µ < k. Proof. From Proposition 5.11.7, G has the M-property if and only if t(k + 1)2 ≤ µk(t + 1). Then, taking into account that tµ, µ ≤ k − 1, we get that h i 1 ≤ t ≤ µkt + µk − tk2 − 2tk ≤ 2k(k − 1) − tk2 − 2tk = k (2 − t)k − 2(t + 1) and hence t = 1; that is, Γ = T (k, µ), and k ≥ 5. In this case T (k, µ) has the M-property if and only if (k + 1)2 ≤ 2µk; that is, if and only if 1 ≤ k(2µ − 2 − k) or, equivalently, 2µ ≥ k + 3.

If G is a Taylor graph with 1 ≤ µ < k − 1, it is well known that the graph Γ2 is also a Taylor graph, whose intersection array is ι(Γ2) = {k, k − 1 − µ, 1; 1, k − 1 − µ, k}. Then, Γ2 has the M-property if and only if µ ≤ (k − 5)/2. Corollary 5.11.11. If G is the Taylor graph T (k, µ) with 1 ≤ µ ≤ k − 2, then either G or Γ2 has the M-property, with the exception of the cases where µ ∈ {m−2, m−1, m, m+1} when k = 2m, or µ ∈ {m−1, m, m+1} when k = 2m+1, in which case none of them have the M-property. Both bipartite and antipodal distance-regular graphs with D = 3 have the intersection array ι(Γ) = {k, k − 1, 1; 1, k − 1, k} and are called k-crown graphs. Therefore, they are Taylor graphs with µ = k − 1 and hence have the M-property if and only if k ≥ 5. We ﬁnish this chapter by studying the primitive case. First, we remark that the characterization given in Proposition 5.11.7 implies that 3k > n which in turn also implies several easy-to-check necessary conditions for G to have the M- property. To obtain them, we need the following special case of the result Brouwer– Cohen–Neumaier [16, Lem. 5.1.2], where we take into account that Taylor graphs are precisely antipodal 2-covers with diameter 3.

Lemma 5.11.12. If G is a distance-regular graph with D = 3, then k2 = k if and only if G is either C6, C7, or T (µ, k). Proposition 5.11.13. If G is a distance-regular graph with D = 3 satisfying the M-property, then 1 < c2 ≤ b1 < 2c2, b2 < c3, and k3 ≤ k − 3. Moreover, k ≥ 6 and c2 < b1 when, in addition, G is not a Taylor graph. Proof. Clearly, G is not a cycle, since it has the M-property. On the other hand, if G is the Taylor graph T (k, µ), then k ≥ 5 and µ ≥ 4 and, moreover, it satisﬁes 5.11. Distance-regular Graphs with the M-Property 213

the inequalities 1 < c2 = b1 < 2c2, 1 = b2 < c3 = k, and 1 = k3 ≤ k −3. Therefore, from Lemma 5.11.12, we can assume that k2 > k, and hence c2 < b1. From the equality n = 1 + k + k2 + k3 we obtain k2 ≤ 2k − 3 and k3 ≤ 2(k − 1) − k2 < k − 2, as otherwise n ≥ 3k and G would not have the M-property. Clearly, the inequality k2 < 2k implies that b1 < 2c2, whereas the inequality k3 ≤ k − 3 < k implies both k ≥ 4 and b1b2 < c2c3 and hence b2 < c3, since c2 < b1. Moreover, c2 > 1, since if c2 = 1, the inequalities 1 = c2 < b1 < 2c2 = 2 are impossible. Finally, when G is not a Taylor graph, from the inequality k ≤ k3(k3 − 1) ≤ (k − 3)(k − 4) we conclude that, necessarily, k ≥ 6. As a byproduct of the above result and applying Brouwer–Cohen–Neumaier [16, Thm. 5.4.1, Prop. 5.4.3], we obtain the strict monotonicity of the intersection parameters of distance-regular graphs with D = 3 and satisfying the M-property. Corollary 5.11.14. If G is a distance-regular graph with diameter 3 that has the M-property, then b2 < b1 < k and 1 < c2 < c3.

The inequalities on the parameters b1, b2, c2, c3 given in Proposition 5.11.13, speciﬁcally the inequality b1 < 2c2, show that none of the families of primitive distance-regular graphs with diameter 3 listed in [12, 16] satisfy the M-property. Furthermore, none of the Shilla graphs recently introduced in Koolen–Park [31] satisfy the M-property. In Bendito–Carmona–Encinas–Mitjana [8] we conjectured that there are no primitive distance-regular graphs with diameter 3 satisfying the M-property. Indeed, Koolen–Park [32, Thm. 1] shows that the conjecture is true except for ﬁnitely many graphs. Bibliography

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