DOCSLIB.ORG

Chapter 3 -- Spectral Graph Theory and Random Walks

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 3 -- Spectral Graph Theory and Random Walks Chapter 3 Spectral graph theory and random walks on graphs Algebraic graph theory is a major area within graph theory. One of the main themes of algebraic graph theory comes from the following question: what do matrices and linear algebra tell us about graphs? One of the most useful invariants of a matrix to look in linear algebra at are its eigenvalues and eigenspaces. This is also true in graph theory, and this aspect of graph theory is known as spectral graph theory. Given a graph G, the most obvious matrix to look at is its adjacency matrix A, however there are others. Two other common choices are the Laplacian matrix, motivated from di↵erential geometry, and what we will call the transition matrix, motivated from dynamical systems—specifically random walks on graphs. The Laplacian and transition matrices are closely related (the eigenvalues of one determine the eigenvalues of the other), and which to look at just depends upon one’s point of view. We will first examine some aspects of the adjacency matrix, and then discuss random walks on graphs and transition matrices. On one hand, eigenvalues can be used to measure how good the network flow is and give bounds on the diameter of a graph. This will help us answer some of the questions we raised in the first chapter about communication and transportation networks. On the other hand, eigenvectors will tell us more precise information about the flow of a networks. One application is using eigenvectors to give a di↵erent kind of centrality ranking, and we will use this idea when we explain Google PageRank. Here are some other references on these topics. Algebraic Graph Theory, by Norman Biggs. • Algebraic Graph Theory, by Chris Godsil and Gordon Royle. • Modern Graph Theory, by B´ela Bollob´as. • Spectra of Graphs, by Andries Brouwer and Willem Haemers. • Spectral Graph Theory, by Fan Chung. • The first two books are “classical graph theory” books in the sense that they do not discuss random walks on graphs, and cover more than just spectral theory. Bollob´as’s book covers many 105 Graph Theory/Social Networks Chapter 3 Kimball Martin (Spring 2014) topics, including some spectral theory and random walks on graphs (and random graphs). The latter two books focus on spectral theory. Brouwer–Haemers cover the adjacency and Laplacian spectra but does not really discuss random walks, whereas Chung’s book discusses random walks but focuses entirely on the (normalized) Laplacian matrix. 3.1 A review of some linear algebra Before we get started on applying some linear algebra to graph theory, we will review some standard facts from linear algebra. This is just meant to be a quick reminder of the relevant theory, as well as fixing some notation, but you should refer to a linear algebra text to make sense of what I’m saying. Let Mn(R) denote the set of real n n matrices, and Mn(C) denote the set of complex n n ⇥ ⇥ matrices. In what follows, we mainly work with real matrices, though most of the theory is the same for complex matrices. Let A Mn(R). We may view A as a linear operator 2 n n A : R R ! v Av 7! where we view v Rn as a n 1 matrix, i.e., a column vector. We may sometimes write a column 2 ⇥ x1 . vector v = 0 . 1 as v =(x1,...,xn) out of typographical convenience. xn t B C t t t t t Let A denote@ A the transpose of A.Then(A ) = A. For A, B Mn(R), we have (AB) = B A . 2 Recall AB does not usually equal BA, though it happens from time to time. Denote by I = In the n n identity matrix. This means AI = IA = A for any A Mn(R). ⇥ 2 We use 0 for the number 0 as well as zero matrices. Denote by diag(d1,...,dn) the diagonal matrix with A =(aij)withaii = di and aij =0if i = j. Note I = diag(1, 1,...,1). 6 Recall A is invertible if there is a matrix B Mn(R) such that AB = I.Ifthereissucha 2 1 matrix B, it must be unique, and we say B is the inverse of A and write B = A− . It is true that 1 1 1 1 1 AA− = A− A = I, i.e., that the inverse of A− is just A,i.e.,(A− )− = A. For A, B Mn(R) 1 1 1 2 both invertible, we have that AB is also invertible and (AB)− = B− A− . There is a function det : Mn(R) R called the determinant. It satisfies det(AB)=det(A)det(B), ! and A is invertible if and only if det A = 0. We can define it inductively on n as follows. For n = 1, 6 put det[a]=a. Now A =(aij) Mn(R)withn>1. Let Aij be the ij cofactor matrix, i.e., the 2 matrix obtained by deleting the i-th row and j-th column. Then n 1 det(A)=a det A a det A + +( 1) − a det A . (3.1) 11 11 − 12 12 ··· − 1n 1n Hence the determinant can be computed by using a cofactor expansion along the top row. It can be computed similarly with a cofactor expansion along any row, and since det(A)=det(At), it can be also computed using a cofactor expansion along any column. (Just remember to be careful of signs—the first sign is 1 if you start along an even numbered row or column.) For n = 2, we have − ab det = ad bc. cd − ✓ ◆ 106 Graph Theory/Social Networks Chapter 3 Kimball Martin (Spring 2014) 2 10 − Example 3.1.1. Let A = 02 1 . Then 0 − 1 10 2 − @ A 2 1 0 1 02 det A =2det − ( 1) det − +0det 02− − 12 10 ✓ ◆ ✓− ◆ ✓− ◆ =2 4+1 ( 1) + 0 = 7. · · − For larger n, determinants are unpleasant to compute by hand in general, but some special cases can be worked out. For instance, if D = diag(d1,...,dn) is diagonal, then det D = d1d2 dn.In n ··· particular, det I = 1, and det kIn = k for a scalar k R. Also note that for invertible A,since 1 1 2 1 1 det(A)det(A− )=det(AA− )=detI = 1, we have det(A− )=(detA)− .IfS is invertible, then 1 1 det(SAS− )=detS det A det S− =detA. Hence similar matrices have the same determinant. Thus it makes sense to define the determinant det T of a linear transformation (i.e., this determinant does not depend upon a choice of basis). n n Given any linear transformation T : R R , and any basis = v1,...vn , we can associate ! B { } a matrix [T ] in Mn(R) such that if B T (x v + x v )=y v + + y v 1 1 ··· n n 1 1 ··· n n then x1 y1 [T ] . = . B 0 . 1 0 . 1 x y B nC B nC @ A @ A n In particular, let 0 = e1,...,en denote the standard basis for R ,i.e.,ei is the column vector B { } with a 1 in the i-th position and 0’s elsewhere. Then, if Tei = ui,thei-th column of [T ] 0 is ui, B i.e., [T ] 0 =[u1 u2 un]. B | |···| Given any bases = v ,...,v and = v ,...,v , there is an invertible change of basis B { 1 n} B0 { 10 n0 } matrix S,determinedbyrequiringSvi = vi0 for all i, such that 1 [T ] = S[T ] S− . (3.2) B0 B 1 For any matrices A, B Mn(R), we say they are similar if A = SAS− for an invertible S Mn(R). 2 2 This means A and B can be viewed as representing the same linear transformation T ,justwith respect to di↵erent bases. We say a nonzero vector v Rn is a (real) eigenvector for A with (real) eigenvalue λ R if 2 2 Av = λv.Ifλ is an eigenvalue for A, the set of v Rn (including the zero vector) for which 2 n Av = λv is called the λ-eigenspace for A, and it is a linear subspace of R .EvenifA Mn(R) 2 it may not have any real eigenvalues/vectors but it may have complex ones, so we will sometimes consider those. The dimension of the λ-eigenspace is called the geometric multiplicity of λ. Eigenvalues and eigenvectors are crucial to understanding the geometry of a linear transfor- mation as A having real eigenvalue λ means A simply acts as scaling by λ on the λ-eigenspace. i✓ Any complex eigenvalues must occur in pairs re± , and complex eigenvalues means that A acts by scaling by r together with rotation by ✓ on an appropriate 2-dimensional subspace. 107 Graph Theory/Social Networks Chapter 3 Kimball Martin (Spring 2014) Eigenvalues can be computed as follows. If Av = λv = λIv for a nonzero v, this means (λI A) − is not invertible, so it has determinant 0. The characteristic polynomial of A is p (λ)=det(λI A). (3.3) A − This is a polynomial of degree n in λ, and its roots are the eigenvalues. By the fundamental theorem of algebra, over C it always factors into linear terms p (λ)=(λ λ )m1 (λ λ )m2 (λ λ )mk . (3.4) A − 1 − 2 ··· − k Here λ1,...,λk denote the distinct (possibly complex) eigenvalues. The number mi is called the (algebraic) multiplicity of λi. It is always greater than or equal to the (complex) dimension of the (complex) λi-eigenspace (the geometric multiplicity).
Load more

Recommended publications
  • Algebraic Graph Theory: Automorphism Groups and Cayley Graphs
    Algebraic Graph Theory: Automorphism Groups and Cayley graphs Glenna Toomey April 2014 1 Introduction An algebraic approach to graph theory can be useful in numerous ways. There is a relatively natural intersection between the fields of algebra and graph theory, specifically between group theory and graphs. Perhaps the most natural connection between group theory and graph theory lies in finding the automorphism group of a given graph. However, by studying the opposite connection, that is, finding a graph of a given group, we can define an extremely important family of vertex-transitive graphs. This paper explores the structure of these graphs and the ways in which we can use groups to explore their properties. 2 Algebraic Graph Theory: The Basics First, let us determine some terminology and examine a few basic elements of graphs. A graph, Γ, is simply a nonempty set of vertices, which we will denote V (Γ), and a set of edges, E(Γ), which consists of two-element subsets of V (Γ). If fu; vg 2 E(Γ), then we say that u and v are adjacent vertices. It is often helpful to view these graphs pictorially, letting the vertices in V (Γ) be nodes and the edges in E(Γ) be lines connecting these nodes. A digraph, D is a nonempty set of vertices, V (D) together with a set of ordered pairs, E(D) of distinct elements from V (D). Thus, given two vertices, u, v, in a digraph, u may be adjacent to v, but v is not necessarily adjacent to u. This relation is represented by arcs instead of basic edges.
    [Show full text]
  • On Cayley Graphs of Algebraic Structures
    On Cayley graphs of algebraic structures Didier Caucal1 1 CNRS, LIGM, University Paris-East, France [email protected] Abstract We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree. 1 Introduction To describe the structure of a group, Cayley introduced in 1878 [7] the concept of graph for any group (G, ·) according to any generating subset S. This is simply the set of labeled s oriented edges g −→ g·s for every g of G and s of S. Such a graph, called Cayley graph, is directed and labeled in S (or an encoding of S by symbols called letters or colors). The study of groups by their Cayley graphs is a main topic of algebraic graph theory [3, 8, 2]. A characterization of unlabeled and undirected Cayley graphs was given by Sabidussi in 1958 [15] : an unlabeled and undirected graph is a Cayley graph if and only if we can find a group with a free and transitive action on the graph. However, this algebraic characterization is not well suited for deciding whether a possibly infinite graph is a Cayley graph. It is pertinent to look for characterizations by graph-theoretic conditions. This approach was clearly stated by Hamkins in 2010: Which graphs are Cayley graphs? [10]. In this paper, we present simple graph-theoretic characterizations of Cayley graphs for firstly left-cancellative and cancellative monoids, and then for groups. These characterizations are then extended to any subset S of left-cancellative magmas, left-quasigroups, quasigroups, and groups.
    [Show full text]
  • An Introduction to Algebraic Graph Theory
    An Introduction to Algebraic Graph Theory Cesar O. Aguilar Department of Mathematics State University of New York at Geneseo Last Update: March 25, 2021 Contents 1 Graphs 1 1.1 What is a graph? ......................... 1 1.1.1 Exercises .......................... 3 1.2 The rudiments of graph theory .................. 4 1.2.1 Exercises .......................... 10 1.3 Permutations ........................... 13 1.3.1 Exercises .......................... 19 1.4 Graph isomorphisms ....................... 21 1.4.1 Exercises .......................... 30 1.5 Special graphs and graph operations .............. 32 1.5.1 Exercises .......................... 37 1.6 Trees ................................ 41 1.6.1 Exercises .......................... 45 2 The Adjacency Matrix 47 2.1 The Adjacency Matrix ...................... 48 2.1.1 Exercises .......................... 53 2.2 The coefficients and roots of a polynomial ........... 55 2.2.1 Exercises .......................... 62 2.3 The characteristic polynomial and spectrum of a graph .... 63 2.3.1 Exercises .......................... 70 2.4 Cospectral graphs ......................... 73 2.4.1 Exercises .......................... 84 3 2.5 Bipartite Graphs ......................... 84 3 Graph Colorings 89 3.1 The basics ............................. 89 3.2 Bounds on the chromatic number ................ 91 3.3 The Chromatic Polynomial .................... 98 3.3.1 Exercises ..........................108 4 Laplacian Matrices 111 4.1 The Laplacian and Signless Laplacian Matrices .........111 4.1.1
    [Show full text]
  • Algebraic Graph Theory
    An Introduction to Algebraic Graph Theory Rob Beezer [email protected] Department of Mathematics and Computer Science University of Puget Sound Mathematics Department Seminar Pacific University October 19, 2009 Labeling Puzzles −1 1 Assign a single real number value to each circle. For each circle, sum the 0 0 values of adjacent circles. Goal: Sum at each circle should be a common multiple of the 1 −1 value at the circle. Common multiple: −2 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 2 / 36 Example Solutions 1 1 1 1 1 1 Common multiple: 3 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 3 / 36 Example Solutions 1 −1 1 −1 1 −1 Common multiple: 1 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 4 / 36 Example Solutions −1 −1 0 0 1 1 Common multiple: 0 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 5 / 36 Example Solutions 0 −1 1 0 Common multiple: −1 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 6 / 36 Example Solutions −1 0 0 1 Common multiple: 0 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 7 / 36 Example Solutions r−1 4 r=Sqrt(17) 4 r−1 p 1 1 Common multiple: 2 (r + 1) = 2 17 + 1 Rob Beezer (U Puget Sound) An Introduction to Algebraic Graph Theory Pacific Math Oct 19 2009 8 / 36 Graphs A graph is a collection of vertices (nodes, dots) where some pairs are joined by edges (arcs, lines).
    [Show full text]
  • New Large Graphs with Given Degree and Diameter ∗
    New Large Graphs with Given Degree and Diameter ∗ F.Comellas and J.G´omez Departament de Matem`aticaAplicada i Telem`atica Universitat Polit`ecnicade Catalunya Abstract In this paper we give graphs with the largest known order for a given degree ∆ and diameter D. The graphs are constructed from Moore bipartite graphs by replacement of some vertices by adequate structures. The paper also contains the latest version of the (∆,D) table for graphs. 1 Introduction A question of special interest in Graph theory is the construction of graphs with an order as large as possible for a given maximum de- gree and diameter or (∆,D)-problem. This problem receives much attention due to its implications in the design of topologies for inter- connection networks and other questions such as the data alignment problem and the description of some cryptographic protocols. The (∆,D) problem for undirected graphs has been approached in different ways. It is possible to give bounds to the order of the graphs for a given maximum degree and diameter (see [5]). On the other hand, as the theoretical bounds are difficult to attain, most of the work deals with the construction of graphs, which for this given diameter and maximum degree have a number of vertices as close as possible to the theoretical bounds (see [4] for a review). Different techniques have been developed depending on the way graphs are generated and their parameters are calculated. Many (∆,D) ∗ Research supported in part by the Comisi´onInterministerial de Ciencia y Tec- nologia, Spain, under grant TIC90-0712 1 graphs correspond to Cayley graphs [6, 11, 7] and have been found by computer search.
    [Show full text]
  • Distance-Transitive Graphs
    Distance-Transitive Graphs Submitted for the module MATH4081 Robert F. Bailey (4MH) Supervisor: Prof. H.D. Macpherson May 10, 2002 2 Robert Bailey Department of Pure Mathematics University of Leeds Leeds, LS2 9JT May 10, 2002 The cover illustration is a diagram of the Biggs-Smith graph, a distance-transitive graph described in section 11.2. Foreword A graph is distance-transitive if, for any two arbitrarily-chosen pairs of vertices at the same distance, there is some automorphism of the graph taking the first pair onto the second. This project studies some of the properties of these graphs, beginning with some relatively simple combinatorial properties (chapter 2), and moving on to dis- cuss more advanced ones, such as the adjacency algebra (chapter 7), and Smith’s Theorem on primitive and imprimitive graphs (chapter 8). We describe four infinite families of distance-transitive graphs, these being the Johnson graphs, odd graphs (chapter 3), Hamming graphs (chapter 5) and Grass- mann graphs (chapter 6). Some group theory used in describing the last two of these families is developed in chapter 4. There is a chapter (chapter 9) on methods for constructing a new graph from an existing one; this concentrates mainly on line graphs and their properties. Finally (chapter 10), we demonstrate some of the ideas used in proving that for a given integer k > 2, there are only finitely many distance-transitive graphs of valency k, concentrating in particular on the cases k = 3 and k = 4. We also (chapter 11) present complete classifications of all distance-transitive graphs with these specific valencies.
    [Show full text]
  • Regular Graphs
    Taylor & Francis I-itt¿'tt¡'tnul "llultilintur ..llsahru- Vol. f4 No.1 1006 i[r-it O T¡yld &kilrr c@p On outindependent subgraphs of strongly regular graphs M. A. FIOL* a¡rd E. GARRIGA l)epartarnent de Matenr¿itica Aplicirda IV. t.luiversit¿rr Politdcnica de Catalunv¡- Jordi Gi'on. I--l- \{t\dul C-1. canrpus Nord. 080-lJ Barcelc¡'a. Sp'i¡r Conrr¡¡¡¡¡.n,"d br W. W¿rtkins I Rr,t t,llr,r/ -aj .llrny'r -rllllj) \n trutl¡dcpe¡dollt suhsr¿tnh ot'a gt'aph f. \\ith respccr to:ul ind!'pcndc¡t vcrtr\ subsct ('C l . rs thc suhl¡.aph f¡-induced bv th!'\ert¡ccs in l'',(', lA'!'s¡ud\ the casc'shcn f is stronslv rcgular. shen: the rcst¡lts ol'de C¿rcn ll99ti. The spc,ctnt ol'conrplcrnenttrn .ubgraph, in qraph. a slron{¡\ rr'!ulitr t-un¡lrt\nt J.rrrnt(l .,1 Ctunl¡in¿!orir:- l9(5)- 559 565.1- allou u: to dc'rivc'thc *lrolc spcctrulrt ol- f¡.. IUorcrrver. rr'lrcn ('rtt¿rins thr. [lollllr¡n Lor,isz bound l'or the indePcndr'nce llu¡rthcr. f( ii a rcsul:rr rr:rph {in lüct. d¡st:rnce-rc.rtular il' I- is I \toore graph). This arliclc'i: nltinlr deltrled to stutll: the non-lcgulal case. As lr nlain result. se chitractcrizc thc structurc ol- i¿^ s ht'n (' is ¡he n..ighhorhood of cirher ¡nc \!.rtc\ ()r ¡ne cdse. Ai,r rrr¡riA. Stlonsh rcuular lra¡rh: Indepcndt-nt set: Sr)ett¡.unr .
    [Show full text]
  • Problems in Algebraic Combinatorics
    PROBLEMS IN ALGEBRAIC COMBINATORICS C. D. Godsil 1 Combinatorics and Optimization University of Waterloo Waterloo, Ontario Canada N2L 3G1 [email protected] Submitted: July 10, 1994; Accepted: January 20, 1995. Abstract: This is a list of open problems, mainly in graph theory and all with an algebraic flavour. Except for 6.1, 7.1 and 12.2 they are either folklore, or are stolen from other people. AMS Classification Number: 05E99 1. Moore Graphs We define a Moore Graph to be a graph with diameter d and girth 2d +1. Somewhat surprisingly, any such graph must necessarily be regular (see [42]) and, given this, it is not hard to show that any Moore graph is distance regular. The complete graphs and odd cycles are trivial examples of Moore graphs. The Petersen and Hoffman-Singleton graphs are non-trivial examples. These examples were found by Hoffman and Singleton [23], where they showed that if X is a k-regular Moore graph with diameter two then k ∈{2, 3, 7, 57}. This immediately raises the following question: 1 Support from grant OGP0093041 of the National Sciences and Engineering Council of Canada is gratefully acknowledged. the electronic journal of combinatorics 2 (1995), #F1 2 1.1 Problem. Is there a regular graph with valency 57, diameter two and girth five? We summarise what is known. Bannai and Ito [4] and, independently, Damerell [12] showed that a Moore graph has diameter at most two. (For an exposition of this, see Chapter 23 in Biggs [7].) Aschbacher [2] proved that a Moore graph with valency 57 could not be distance transitive and G.
    [Show full text]
  • From Static to Dynamic Couplings in Consensus and Synchronization Among Identical and Non-Identical Systems
    From Static to Dynamic Couplings in Consensus and Synchronization among Identical and Non-Identical Systems Von der Fakult¨at Konstruktions-, Produktions-, und Fahrzeugtechnik der Universit¨at Stuttgart zur Erlangung der Wurde¨ eines Doktors der Ingenieurwissenschaften (Dr.–Ing.) genehmigte Abhandlung Vorgelegt von Peter Wieland aus Stuttgart–Bad Cannstatt Hauptberichter: Prof. Dr.–Ing. Frank Allg¨ower Mitberichter: Prof. Dr. Alberto Isidori Prof. Dr. Rodolphe Sepulchre Tag der mundlichen¨ Prufung:¨ 06.09.2010 Institut fur¨ Systemtheorie und Regelungstechnik Universit¨at Stuttgart 2010 F¨ur Melanie, Lea und Anna Acknowledgments The results presented in this thesis are the outcome of the research I performed during my time as a research assistant at the Institute for Systems Theory and Automatic Control (IST), University of Stuttgart in the years 2005 to 2010. Throughout this exciting journey I had the chance to meet and interact with many interesting and inspiring people. The present thesis was positively influenced by many of them and I wish to convey my sincere gratefulness to them. First, I want to express my gratitude towards my supervisor, Prof. Frank Allg¨ower. He endowed me with lots of freedom in my research and offered me the unique opportunity to be part of a very open-minded, internationally renowned research group. I also want to thank Prof. Rodolphe Sepulchre, with whom I had the chance to have a very fruitful interaction during the last year of my thesis, for sharing his knowledge and insight with me in a very motivating way. Moreover, I am deeply indebted to Prof. Alberto Isidori, Prof. Rodolphe Sepulchre, and Prof.
    [Show full text]
  • SM444 Notes on Algebraic Graph Theory
    SM444 notes on algebraic graph theory David Joyner 2017-12-04 These are notes1 on algebraic graph theory for sm444. This course focuses on \calculus on graphs" and will introduce and study the graph-theoretic analog of (for example) the gradient. For some well-made short videos on graph theory, I recommend Sarada Herke's channel on youtube. Contents 1 Introduction 2 2 Examples 6 3 Basic definitions 20 3.1 Diameter, radius, and all that . 20 3.2 Treks, trails, paths . 22 3.3 Maps between graphs . 26 3.4 Colorings . 29 3.5 Transitivity . 30 4 Adjacency matrix 31 4.1 Definition . 31 4.2 Basic results . 32 4.3 The spectrum . 35 4.4 Application to the Friendship Theorem . 44 4.5 Eigenvector centrality . 47 1See Biggs [Bi93], Joyner-Melles [JM], Joyner-Phillips [JP] and the lecture notes of Prof Griffin [Gr17] for details. 1 4.5.1 Keener ranking . 52 4.6 Strongly regular graphs . 54 4.6.1 The Petersen graph . 54 4.7 Desargues graph . 55 4.8 D¨urergraph . 56 4.9 Orientation on a graph . 57 5 Incidence matrix 59 5.1 The unsigned incidence matrix . 59 5.2 The oriented case . 62 5.3 Cycle space and cut space . 63 6 Laplacian matrix 74 6.1 The Laplacian spectrum . 79 7 Hodge decomposition for graphs 84 7.1 Abstract simplicial complexes . 85 7.2 The Bj¨orner complex and the Riemann hypothesis . 91 7.3 Homology groups . 95 8 Comparison graphs 98 8.1 Comparison matrices . 98 8.2 HodgeRank . 100 8.3 HodgeRank example .
    [Show full text]
  • Spectral Graph Theory to Appear in Handbook of Linear Algebra, Second Edition, CCR Press
    Spectral Graph Theory to appear in Handbook of Linear Algebra, second edition, CCR Press Steve Butler∗ Fan Chungy There are many different ways to associate a matrix with a graph (an introduction of which can be found in Chapter 28 on Matrices and Graphs). The focus of spectral graph theory is to examine the eigenvalues (or spectrum) of such a matrix and use them to determine structural properties of the graph; or conversely, given a structural property of the graph determine the nature of the eigenvalues of the matrix. Some of the different matrices we can use for a graph G include the adjacency matrix AG, the (combina- torial) Laplacian matrix LG, the signless Laplacian matrix jLGj, and the normalized Laplacian matrix LG. The eigenvalues of these different matrices can be used to find out different information about the graph. No one matrix is best because each matrix has its own limitations in that there is some property which the matrix cannot always determine (this shows the importance of understanding all the different matrices). In the table below we summarize the four matrices and four different structural properties of a graph and indicate which properties can be determined by the eigenvalues. A \No" answer indicates the existence of two non-isomorphic graphs which have the same spectrum but differ in the indicated structure. Matrix bipartite # components # bipartite components # edges Adjacency | AG Yes No No Yes (combinatorial) Laplacian | LG No Yes No Yes Signless Laplacian | jLGj No No Yes Yes Normalized Laplacian | LG Yes Yes Yes No In this chapter we will introduce the normalized Laplacian and give some basic properties of this matrix and some additional applications of the eigenvalues of graphs.
    [Show full text]
  • Algebraic Graph Theory: Including Cayley Graphs, Group Actions On
    Algebraic graph theory: including Cayley graphs, group actions on graphs, graph eigenvalues, graphs and matrices Th´eorie des graphes alg´ebrique: graphes de Cayley, actions de groupe sur les graphes, valeurs propres d’un graphe, graphes et matrices (Org: Joy Morris (Lethbridge)) ROBERT BAILEY, Grenfell Campus, Memorial University On the metric dimension of incidence graphs A resolving set for a graph Γ is a collection of vertices chosen so that any vertex of Γ is uniquely identified by the list of distances to the chosen few. The metric dimension of Γ is the smallest size of a resolving set for Γ. In this talk, we will consider the incidence graphs of symmetric designs, and show how he probabilistic method can be used to bound their metric dimension. In the case of incidence graphs of Hadamard designs, this result is (asymptotically) best possible. JOEL FRIEDMAN, University of British Columbia Sheaves on Graphs and Applications We will give an introduction to sheaves on graphs and briefly discuss some of their applications. A sheaf of vector spaces on a graph is a family of vector spaces, one for each vertex and one for each edge, with maps from each edge’s space to those of its incident vertices. Each sheaf has its own incidence matrix, Laplacians, adjecency matrices, etc. Sheaves allow us to (1) compare different sheaves over the same graph via exact sequences, and (2) create ”new” morphisms between graphs, e.g., when there is no surjection of one graph to another, there can be surjections of the sheaves ”representing” the graphs.
    [Show full text]