Alternating Sign Matrices, Related (0,1)-Matrices, and the Smith

Total Page:16

File Type:pdf, Size:1020Kb

Alternating Sign Matrices, Related (0,1)-Matrices, and the Smith Alternating Sign Matrices, Related (0; 1)-Matrices, and the Smith Normal Form Richard A. Brualdi∗ Geir Dahly March 11, 2018 Dedicated with admiration to Vladimir Sergeichuk on the occasion of his 70th birthday. Abstract We investigate the Smith Normal Form (SNF) of alternating sign matrices (ASMs) and related matrices of 0's and 1's ((0; 1)-matrices). We identify certain classes of ASMs and (0; 1)-matrices whose SNFs are (0; 1)-matrices. We relate some of our work to various ranks, in particular, the (0; 1)-rank of a (0; 1)-matrix, that is, the bipartite partition number of a bipartite graph. Key words. Alternating sign matrix, convex (0; 1)-matrix, rank, bipartite graph, Smith normal form. AMS subject classifications. 05B20, 15B48. 1 Introduction An alternating sign matrix, abbreviated to ASM, is an n × n (0; ±1)-matrix such that, ignoring 0's, the 1's and −1's in each row and column alternate beginning ∗Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. [email protected] yDepartment of Mathematics, University of Oslo, Norway. [email protected]. Correspond- ing author. 1 and ending with a 1. ASMs can be regarded as generalizations of permutation matrices, since permutation matrices are the ASMs without any −1's. Some ASMs A containing −1's are signature equivalent to (0; 1)-matrices, that is, there may exist diagonal matrices D1 and D2 all of whose diagonal entries are 1 or −1, such that D1AD2 is a (0; 1)-matrix. The matrix D1AD2 results from A by multiplying certain rows and columns by −1. The following example illustrates these observations. Example 1.1. 1 −1 1 −1 2 1 3 −1 1 6 1 1 1 7 1 1 −1 1 ! 6 7 : 4 1 1 1 5 −1 1 −1 1 1 1 1 Recall the three basic unimodular row operations on an integral matrix: (i) interchanging two rows, (ii) multiplying a row by −1, and (iii) adding an integral multiple of one row to another row. Operations (i), (ii) and (iii) are called elementary row operations, abbreviated to EROs. Unimodular column operations are defined similarly and we get ECOs. For simplicity, we refer to both of these operations as elementary operations or EOs. Thus D1AD2 results from A by a sequence of EOs of type (ii). Let A be an m × n integral matrix. Then, by EOs, A can be brought to diagonal form 2 3 e1 0 ··· 0 6 0 e2 ··· 0 Or;n−r 7 6 7 6 . .. 7 6 . 7 6 7 (1) 6 0 0 ··· er 7 6 7 4 5 Om−r;r Om−r;n−r where r is the rank of A, and e1; e2; : : : ; er are positive integers satisfying the divis- ibility conditions e1je2; e2je3; : : : ; er−1jer. The matrix (1) is called the Smith normal form of A, denoted as SNF(A), and e1; e2; : : : ; er are the invariant factors of A. The invariant factors of A are unique and, in fact, ei = di=di−1 where di is the greatest common divisor of all determinantal minors of size i (1 ≤ i ≤ r) and d0 = 1. The positive integers d1; d2; : : : ; dr are the elementary divisors of A. If A is nonsingular, then r = n and e1e2 ··· en = j det(A)j. Note that the SNF of a matrix is, in partic- ular, invariant under permutations of its rows and of its columns. The paper [18] surveys some aspects of the Smith normal form of combinatorially defined matrices. 2 We say that A has a (0; 1)-SNF provided that its invariant factors all equal 1, that is, provided I O SNF(A) = r (2) OO where r is the rank of A. Neither the Smith normal form of an ASM nor of a (0; 1)-matrix needs to be a (0; 1)-matrix as shown in the following examples. Example 1.2. We have 02 1 1 0 31 2 1 0 0 3 SNF @4 0 1 1 5A = 4 0 1 0 5 : (3) 1 0 1 0 0 2 This follows by observing that this matrix has determinant equal to 2 and all 2 × 2 minors have value ±1. Now consider the following 12 × 12 ASM A where we use + to denote a 1 and − to denote a −1: 2 + 3 6 + 7 6 7 6 + 7 6 7 6 + − + 7 6 7 6 + − + 7 6 7 6 7 6 + − + 7 A = 6 7 : 6 + − + 7 6 7 6 + − + 7 6 7 6 + − + 7 6 7 6 + 7 6 7 4 + 5 + The shaded entries of A indicate a cycle configuration of consecutive entries of the form (+ − +) such that any two have at most one entry in common. To find the SNF of A we use each of the 1's in the first three rows and eliminate the nonzeros below, and then we use each of the 1's in the first three columns and eliminate the 3 nonzeros to the right. The resulting matrix is 2 + 3 6 + 7 6 7 6 + 7 6 7 6 + + 7 6 7 6 + 7 6 7 6 7 6 + + 7 B = 6 7 : 6 + 7 6 7 6 + 7 6 7 6 + + 7 6 7 6 7 6 7 4 5 Note the 6 ones in shaded cells now form a bipartite cycle. Permuting rows and columns so that this cycle corresponds to rows and columns 7; 8; 9 we get 2 1 3 6 1 7 6 7 6 1 7 6 7 6 1 7 6 7 6 1 7 6 7 6 7 0 6 1 7 B = 6 7 : 6 1 1 7 6 7 6 1 1 7 6 7 6 1 1 7 6 7 6 7 6 7 4 5 Then (see (3)) SNF(B0) = SNF(A) is the diagonal matrix with diagonal (1; 1; 1; 1; 1; 1; 1; 1; 2; 0; 0; 0); and hence A does not have a (0; 1)-SNF. If an n×n integral matrix A satisfies that, for k = 1; 2; : : : ; n, A contains a minor which is ±1, then it follows that every elementary divisor, and hence every invariant factor, is 1, so that A has a (0; 1)-SNF. ASMs satisfying this sufficient condition may have some interesting properties. 4 Our goal in this paper is to initiate an investigation of the Smith Normal Form of ASMs and related (0; 1)-matrices. Note that a (0; ±1)-matrix A is signature equivalent to a (0; 1)-matrix if and only if there exist permutation matrices P and Q such that A −A P AQ = 11 12 (4) −A21 A22 where A11;A12;A21;A22 are all (0; 1)-matrices. The matrix A22, and hence A12 and A21, may be empty. With a (0; ±1)-matrix A we can associate a bipartite graph with vertex bipartition U (corresponding to the rows) and W (corresponding to the columns), and edges signed as +1 or −1, to get a signed bipartite graph G±(A). The matrix A is then the signed adjacency matrix of G±(A). A cycle of G±(A) is a positive cycle provided the product of the signs of its edges is +1, that is, has an even number of edges with sign −1, and is a negative cycle otherwise. It follows easily that if A is signature equivalent to a (0; 1)-matrix (see (4)), then all cycles of G±(A) are positive. The following known theorem [9, 13] asserts that the converse holds as well. Theorem 1.3. Let A be an m × n (0; ±1)-matrix. Then A is signature equivalent to a (0; 1)-matrix if and only if all cycles of G±(A) are positive. Proof. For the convenience of the reader we outline a brief proof of the converse. So suppose that all cycles of G±(A) are positive, that is, contain an even number of minus edges. Let G+ be the bipartite graph obtained from G± by deleting its minus edges. Then there does not exist a negative edge of G joining two vertices in the same connected component of G+, since such an edge would give a cycle of G with exactly one minus edge. Hence all the minus edges of G, if any, join two vertices in different components of G+. Let H be the graph whose vertices correspond to the connected components of G+ with an edge between two connected components if there is a negative edge between them in G±. Then H is a bipartite graph for, otherwise, H contains a cycle of odd length and then G± contains a negative cycle. The bipartition of the vertices of H induces the partition as given in (4). 2 Convex (0; 1)-Matrices and ASM's Let A = [aij] be a nonzero (0; ±1)-matrix. We say that A is row-convex if there does not exist a zero between two nonzeros in any row. The notion of column-convex is defined similarly. If A is both row- and column-convex, we say that A is convex. A (0; 1)-matrix which is either row-convex or column-convex is usually called an interval matrix. Row-convexity or column-convexity of a (0; 1)-matrix implies a (0; 1)-SNF. 5 Theorem 2.1. Let A be an m × n (0; 1)-matrix which is row-convex or column- convex. Then A has a (0; 1)-SNF. Proof. Assume that A is column-convex. Then A is an interval matrix, and this implies that A is a totally unimodular matrix [14].
Recommended publications
  • Perron–Frobenius Theory of Nonnegative Matrices
    Buy from AMAZON.com http://www.amazon.com/exec/obidos/ASIN/0898714540 CHAPTER 8 Perron–Frobenius Theory of Nonnegative Matrices 8.1 INTRODUCTION m×n A ∈ is said to be a nonnegative matrix whenever each aij ≥ 0, and this is denoted by writing A ≥ 0. In general, A ≥ B means that each aij ≥ bij. Similarly, A is a positive matrix when each aij > 0, and this is denoted by writing A > 0. More generally, A > B means that each aij >bij. Applications abound with nonnegative and positive matrices. In fact, many of the applications considered in this text involve nonnegative matrices. For example, the connectivity matrix C in Example 3.5.2 (p. 100) is nonnegative. The discrete Laplacian L from Example 7.6.2 (p. 563) leads to a nonnegative matrix because (4I − L) ≥ 0. The matrix eAt that defines the solution of the system of differential equations in the mixing problem of Example 7.9.7 (p. 610) is nonnegative for all t ≥ 0. And the system of difference equations p(k)=Ap(k − 1) resulting from the shell game of Example 7.10.8 (p. 635) has Please report violations to [email protected] a nonnegativeCOPYRIGHTED coefficient matrix A. Since nonnegative matrices are pervasive, it’s natural to investigate their properties, and that’s the purpose of this chapter. A primary issue concerns It is illegal to print, duplicate, or distribute this material the extent to which the properties A > 0 or A ≥ 0 translate to spectral properties—e.g., to what extent does A have positive (or nonnegative) eigen- values and eigenvectors? The topic is called the “Perron–Frobenius theory” because it evolved from the contributions of the German mathematicians Oskar (or Oscar) Perron 89 and 89 Oskar Perron (1880–1975) originally set out to fulfill his father’s wishes to be in the family busi- Buy online from SIAM Copyright c 2000 SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 662 Chapter 8 Perron–Frobenius Theory of Nonnegative Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 Ferdinand Georg Frobenius.
    [Show full text]
  • Adjacency and Incidence Matrices
    Adjacency and Incidence Matrices 1 / 10 The Incidence Matrix of a Graph Definition Let G = (V ; E) be a graph where V = f1; 2;:::; ng and E = fe1; e2;:::; emg. The incidence matrix of G is an n × m matrix B = (bik ), where each row corresponds to a vertex and each column corresponds to an edge such that if ek is an edge between i and j, then all elements of column k are 0 except bik = bjk = 1. 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 2 / 10 The First Theorem of Graph Theory Theorem If G is a multigraph with no loops and m edges, the sum of the degrees of all the vertices of G is 2m. Corollary The number of odd vertices in a loopless multigraph is even. 3 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 4 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G.
    [Show full text]
  • Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A
    1 Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A. Deri, Member, IEEE, and José M. F. Moura, Fellow, IEEE Abstract—We define and discuss the utility of two equiv- Consider a graph G = G(A) with adjacency matrix alence graph classes over which a spectral projector-based A 2 CN×N with k ≤ N distinct eigenvalues and Jordan graph Fourier transform is equivalent: isomorphic equiv- decomposition A = VJV −1. The associated Jordan alence classes and Jordan equivalence classes. Isomorphic equivalence classes show that the transform is equivalent subspaces of A are Jij, i = 1; : : : k, j = 1; : : : ; gi, up to a permutation on the node labels. Jordan equivalence where gi is the geometric multiplicity of eigenvalue 휆i, classes permit identical transforms over graphs of noniden- or the dimension of the kernel of A − 휆iI. The signal tical topologies and allow a basis-invariant characterization space S can be uniquely decomposed by the Jordan of total variation orderings of the spectral components. subspaces (see [13], [14] and Section II). For a graph Methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed. signal s 2 S, the graph Fourier transform (GFT) of [12] is defined as Index Terms—Jordan decomposition, generalized k gi eigenspaces, directed graphs, graph equivalence classes, M M graph isomorphism, signal processing on graphs, networks F : S! Jij i=1 j=1 s ! (s ;:::; s ;:::; s ;:::; s ) ; (1) b11 b1g1 bk1 bkgk I. INTRODUCTION where sij is the (oblique) projection of s onto the Jordan subspace Jij parallel to SnJij.
    [Show full text]
  • Network Properties Revealed Through Matrix Functions 697
    SIAM REVIEW c 2010 Society for Industrial and Applied Mathematics Vol. 52, No. 4, pp. 696–714 Network Properties Revealed ∗ through Matrix Functions † Ernesto Estrada Desmond J. Higham‡ Abstract. The emerging field of network science deals with the tasks of modeling, comparing, and summarizing large data sets that describe complex interactions. Because pairwise affinity data can be stored in a two-dimensional array, graph theory and applied linear algebra provide extremely useful tools. Here, we focus on the general concepts of centrality, com- municability,andbetweenness, each of which quantifies important features in a network. Some recent work in the mathematical physics literature has shown that the exponential of a network’s adjacency matrix can be used as the basis for defining and computing specific versions of these measures. We introduce here a general class of measures based on matrix functions, and show that a particular case involving a matrix resolvent arises naturally from graph-theoretic arguments. We also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering. We finish with computational examples showing the new matrix resolvent version applied to real networks. Key words. centrality measures, clustering methods, communicability, Estrada index, Fiedler vector, graph Laplacian, graph spectrum, power series, resolvent AMS subject classifications. 05C50, 05C82, 91D30 DOI. 10.1137/090761070 1. Motivation. 1.1. Introduction. Connections are important. Across the natural, technologi- cal, and social sciences it often makes sense to focus on the pattern of interactions be- tween individual components in a system [1, 8, 61].
    [Show full text]
  • 4.6 Matrices Dimensions: Number of Rows and Columns a Is a 2 X 3 Matrix
    Matrices are rectangular arrangements of data that are used to represent information in tabular form. 1 0 4 A = 3 - 6 8 a23 4.6 Matrices Dimensions: number of rows and columns A is a 2 x 3 matrix Elements of a matrix A are denoted by aij. a23 = 8 1 2 Data about many kinds of problems can often be represented by Matrix of coefficients matrix. Solutions to many problems can be obtained by solving e.g Average temperatures in 3 different cities for each month: systems of linear equations. For example, the constraints of a problem are represented by the system of linear equations 23 26 38 47 58 71 78 77 69 55 39 33 x + y = 70 A = 14 21 33 38 44 57 61 59 49 38 25 21 3 cities 24x + 14y = 1180 35 46 54 67 78 86 91 94 89 75 62 51 12 months Jan - Dec 1 1 rd The matrix A = Average temp. in the 3 24 14 city in July, a37, is 91. is the matrix of coefficient for this system of linear equations. 3 4 In a matrix, the arrangement of the entries is significant. Square Matrix is a matrix in which the number of Therefore, for two matrices to be equal they must have rows equals the number of columns. the same dimensions and the same entries in each location. •Main Diagonal: in a n x n square matrix, the elements a , a , a , …, a form the main Example: Let 11 22 33 nn diagonal of the matrix.
    [Show full text]
  • Recent Developments in Boolean Matrix Factorization
    Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence (IJCAI-20) Survey Track Recent Developments in Boolean Matrix Factorization Pauli Miettinen1∗ and Stefan Neumann2y 1University of Eastern Finland, Kuopio, Finland 2University of Vienna, Vienna, Austria pauli.miettinen@uef.fi, [email protected] Abstract analysts, and van den Broeck and Darwiche [2013] to lifted inference community, to name but a few examples. Even more The goal of Boolean Matrix Factorization (BMF) is recently, Ravanbakhsh et al. [2016] studied the problem in to approximate a given binary matrix as the product the framework of machine learning, increasing the interest of of two low-rank binary factor matrices, where the that community, and Chandran et al. [2016]; Ban et al. [2019]; product of the factor matrices is computed under Fomin et al. [2019] (re-)launched the interest in the problem the Boolean algebra. While the problem is computa- in the theory community. tionally hard, it is also attractive because the binary The various fields in which BMF is used are connected by nature of the factor matrices makes them highly in- the desire to “keep the data binary”. This can be because terpretable. In the last decade, BMF has received a the factorization is used as a pre-processing step, and the considerable amount of attention in the data mining subsequent methods require binary input, or because binary and formal concept analysis communities and, more matrices are more interpretable in the application domain. In recently, the machine learning and the theory com- the latter case, Boolean algebra is often preferred over the field munities also started studying BMF.
    [Show full text]
  • Diagonal Sums of Doubly Stochastic Matrices Arxiv:2101.04143V1 [Math
    Diagonal Sums of Doubly Stochastic Matrices Richard A. Brualdi∗ Geir Dahl† 7 December 2020 Abstract Let Ωn denote the class of n × n doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in Ωn. The main question is: which A 2 Ωn are such that the diagonals in A that avoid the zeros of A all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices. Key words. Doubly stochastic matrix, diagonal sum, patterns. AMS subject classifications. 05C50, 15A15. 1 Introduction Let Mn denote the (vector) space of real n×n matrices and on this space we consider arXiv:2101.04143v1 [math.CO] 11 Jan 2021 P the usual scalar product A · B = i;j aijbij for A; B 2 Mn, A = [aij], B = [bij]. A permutation σ = (k1; k2; : : : ; kn) of f1; 2; : : : ; ng can be identified with an n × n permutation matrix P = Pσ = [pij] by defining pij = 1, if j = ki, and pij = 0, otherwise. If X = [xij] is an n × n matrix, the entries of X in the positions of X in ∗Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. [email protected] †Department of Mathematics, University of Oslo, Norway. [email protected]. Correspond- ing author. 1 which P has a 1 is the diagonal Dσ of X corresponding to σ and P , and their sum n X dP (X) = xi;ki i=1 is a diagonal sum of X.
    [Show full text]
  • The Generalized Dedekind Determinant
    Contemporary Mathematics Volume 655, 2015 http://dx.doi.org/10.1090/conm/655/13232 The Generalized Dedekind Determinant M. Ram Murty and Kaneenika Sinha Abstract. The aim of this note is to calculate the determinants of certain matrices which arise in three different settings, namely from characters on finite abelian groups, zeta functions on lattices and Fourier coefficients of normalized Hecke eigenforms. Seemingly disparate, these results arise from a common framework suggested by elementary linear algebra. 1. Introduction The purpose of this note is three-fold. We prove three seemingly disparate results about matrices which arise in three different settings, namely from charac- ters on finite abelian groups, zeta functions on lattices and Fourier coefficients of normalized Hecke eigenforms. In this section, we state these theorems. In Section 2, we state a lemma from elementary linear algebra, which lies at the heart of our three theorems. A detailed discussion and proofs of the theorems appear in Sections 3, 4 and 5. In what follows below, for any n × n matrix A and for 1 ≤ i, j ≤ n, Ai,j or (A)i,j will denote the (i, j)-th entry of A. A diagonal matrix with diagonal entries y1,y2, ...yn will be denoted as diag (y1,y2, ...yn). Theorem 1.1. Let G = {x1,x2,...xn} be a finite abelian group and let f : G → C be a complex-valued function on G. Let F be an n × n matrix defined by F −1 i,j = f(xi xj). For a character χ on G, (that is, a homomorphism of G into the multiplicative group of the field C of complex numbers), we define Sχ := f(s)χ(s).
    [Show full text]
  • Dynamical Systems Associated with Adjacency Matrices
    DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2018190 DYNAMICAL SYSTEMS SERIES B Volume 23, Number 5, July 2018 pp. 1945{1973 DYNAMICAL SYSTEMS ASSOCIATED WITH ADJACENCY MATRICES Delio Mugnolo Delio Mugnolo, Lehrgebiet Analysis, Fakult¨atMathematik und Informatik FernUniversit¨atin Hagen, D-58084 Hagen, Germany Abstract. We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss in detail qualitative properties of solutions to these problems by quadratic form methods. We distinguish between backward and forward evo- lution equations: the latter have typical features of diffusive processes, but cannot be well-posed on graphs with unbounded degree. On the contrary, well-posedness of backward equations is a typical feature of line graphs. We suggest how to detect even cycles and/or couples of odd cycles on graphs by studying backward equations for the adjacency matrix on their line graph. 1. Introduction. The aim of this paper is to discuss the properties of the linear dynamical system du (t) = Au(t) (1.1) dt where A is the adjacency matrix of a graph G with vertex set V and edge set E. Of course, in the case of finite graphs A is a bounded linear operator on the finite di- mensional Hilbert space CjVj. Hence the solution is given by the exponential matrix etA, but computing it is in general a hard task, since A carries little structure and can be a sparse or dense matrix, depending on the underlying graph.
    [Show full text]
  • A Complete Bibliography of Publications in Linear Algebra and Its Applications: 2010–2019
    A Complete Bibliography of Publications in Linear Algebra and its Applications: 2010{2019 Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 12 March 2021 Version 1.74 Title word cross-reference KY14, Rim12, Rud12, YHH12, vdH14]. 24 [KAAK11]. 2n − 3[BCS10,ˇ Hil13]. 2 × 2 [CGRVC13, CGSCZ10, CM14, DW11, DMS10, JK11, KJK13, MSvW12, Yan14]. (−1; 1) [AAFG12].´ (0; 1) 2 × 2 × 2 [Ber13b]. 3 [BBS12b, NP10, Ghe14a]. (2; 2; 0) [BZWL13, Bre14, CILL12, CKAC14, Fri12, [CI13, PH12]. (A; B) [PP13b]. (α, β) GOvdD14, GX12a, Kal13b, KK14, YHH12]. [HW11, HZM10]. (C; λ; µ)[dMR12].(`; m) p 3n2 − 2 2n3=2 − 3n [MR13]. [DFG10]. (H; m)[BOZ10].(κ, τ) p 3n2 − 2 2n3=23n [MR14a]. 3 × 3 [CSZ10, CR10c]. (λ, 2) [BBS12b]. (m; s; 0) [Dru14, GLZ14, Sev14]. 3 × 3 × 2 [Ber13b]. [GH13b]. (n − 3) [CGO10]. (n − 3; 2; 1) 3 × 3 × 3 [BH13b]. 4 [Ban13a, BDK11, [CCGR13]. (!) [CL12a]. (P; R)[KNS14]. BZ12b, CK13a, FP14, NSW13, Nor14]. 4 × 4 (R; S )[Tre12].−1 [LZG14]. 0 [AKZ13, σ [CJR11]. 5 Ano12-30, CGGS13, DLMZ14, Wu10a]. 1 [BH13b, CHY12, KRH14, Kol13, MW14a]. [Ano12-30, AHL+14, CGGS13, GM14, 5 × 5 [BAD09, DA10, Hil12a, Spe11]. 5 × n Kal13b, LM12, Wu10a]. 1=n [CNPP12]. [CJR11]. 6 1 <t<2 [Seo14]. 2 [AIS14, AM14, AKA13, [DK13c, DK11, DK12a, DK13b, Kar11a].
    [Show full text]
  • Adjacency Matrices Text Reference: Section 2.1, P
    Adjacency Matrices Text Reference: Section 2.1, p. 114 The purpose of this set of exercises is to show how powers of a matrix may be used to investigate graphs. Special attention is paid to airline route maps as examples of graphs. In order to study graphs, the notion of graph must first be defined. A graph is a set of points (called vertices,ornodes) and a set of lines called edges connecting some pairs of vertices. Two vertices connected by an edge are said to be adjacent. Consider the graph in Figure 1. Notice that two vertices may be connected by more than one edge (A and B are connected by 2 distinct edges), that a vertex need not be connected to any other vertex (D), and that a vertex may be connected to itself (F). Another example of a graph is the route map that most airlines (or railways) produce. A copy of the northern route map for Cape Air from May 2001 is Figure 2. This map is available at the web page: www.flycapeair.com. Here the vertices are the cities to which Cape Air flies, and two vertices are connected if a direct flight flies between them. Figure 2: Northern Route Map for Cape Air -- May 2001 Application Project: Adjacency Matrices Page 2 of 5 Some natural questions arise about graphs. It might be important to know if two vertices are connected by a sequence of two edges, even if they are not connected by a single edge. In Figure 1, A and C are connected by a two-edge sequence (actually, there are four distinct ways to go from A to C in two steps).
    [Show full text]
  • COMPUTING RELATIVELY LARGE ALGEBRAIC STRUCTURES by AUTOMATED THEORY EXPLORATION By
    COMPUTING RELATIVELY LARGE ALGEBRAIC STRUCTURES BY AUTOMATED THEORY EXPLORATION by QURATUL-AIN MAHESAR A thesis submitted to The University of Birmingham for the degree of DOCTOR OF PHILOSOPHY School of Computer Science College of Engineering and Physical Sciences The University of Birmingham March 2014 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract Automated reasoning technology provides means for inference in a formal context via a multitude of disparate reasoning techniques. Combining different techniques not only increases the effectiveness of single systems but also provides a more powerful approach to solving hard problems. Consequently combined reasoning systems have been successfully employed to solve non-trivial mathematical problems in combinatorially rich domains that are intractable by traditional mathematical means. Nevertheless, the lack of domain specific knowledge often limits the effectiveness of these systems. In this thesis we investigate how the combination of diverse reasoning techniques can be employed to pre-compute additional knowledge to enable mathematical discovery in finite and potentially infinite domains that is otherwise not feasible. In particular, we demonstrate how we can exploit bespoke symbolic computations and automated theorem proving to automatically compute and evolve the structural knowledge of small size finite structures in the algebraic theory of quasigroups.
    [Show full text]