Adjacency and Incidence Matrices

Total Page：16

File Type：pdf, Size：1020Kb

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. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 1 03 f h 61 0 0 17 3 B = 6 7 g 40 1 0 15 4 0 0 1 0 5 / 10 The Adjacency Matrix of a Graph Definition Let G = (V ; E) be a graph with no multiple edges where V = f1; 2;:::; ng. The adjacency matrix of G is the n × n matrix A = (aij ), where aij = 1 if there is an edge between vertex i and vertex j and aij = 0 otherwise. Notes The adjacency matrix of a graph is symmetric. 6 / 10 Adjacency Matrix Example 1 2 e 20 1 1 13 f 61 0 0 07 3 A = 6 7 g 41 0 0 05 4 1 0 0 0 7 / 10 Vertex Degree Definitions The degree of a vertex in a graph is the number of edges incident on that vertex. A vertex is odd if its degree is odd; otherwise, it is even. Notes The sum of the elements of row i of the adjacency matrix of a graph is the degree of vertex i. The sum of the elements of column i of the adjaceny matrix of a graph is the degree of vertex i. 8 / 10 Linear Algebra and Adjacency Matrices of Graphs Proposition Let A be the adjacency matrix of a graph. The (i; i)-entry in A2 is the degree of vertex i. Recall that the trace of a square matrix is the sum of its diagonal entries. Proposition Let G be a graph with e edges and t triangles. If A is the adjacency matrix of G, then (a) trace(A) = 0, (b) trace(A2) = 2e, (c) trace(A3) = 6t. 9 / 10 Acknowledgements Statements of definitions follow the notation and wording of Balakrishnan's Introductory Discrete Mathematics. 10 / 10.

Copy Link

Recommended publications

Quiz7 Problem 1
Quiz 7 Quiz7 Problem 1. Independence The Problem. In the parts below, cite which tests apply to decide on independence or dependence. Choose one test and show complete details. 1 ! 1 ! (a) Vectors ~v = ;~v = 1 −1 2 1 0 1 1 0 1 1 0 1 1 B C B C B C (b) Vectors ~v1 = @ −1 A ;~v2 = @ 1 A ;~v3 = @ 0 A 0 −1 −1 2 5 (c) Vectors ~v1;~v2;~v3 are data packages constructed from the equations y = x ; y = x ; y = x10 on (−∞; 1). (d) Vectors ~v1;~v2;~v3 are data packages constructed from the equations y = 1 + x; y = 1 − x; y = x3 on (−∞; 1). Basic Test. To show three vectors are independent, form the system of equations c1~v1 + c2~v2 + c3~v3 = ~0; then solve for c1; c2; c3. If the only solution possible is c1 = c2 = c3 = 0, then the vectors are independent. Linear Combination Test. A list of vectors is independent if and only if each vector in the list is not a linear combination of the remaining vectors, and each vector in the list is not zero. Subset Test. Any nonvoid subset of an independent set is independent. The basic test leads to three quick independence tests for column vectors. All tests use the augmented matrix A of the vectors, which for 3 vectors is A =< ~v1j~v2j~v3 >. Rank Test. The vectors are independent if and only if rank(A) equals the number of vectors. Determinant Test. Assume A is square. The vectors are independent if and only if the deter- minant of A is nonzero.
[Show full text]
Directed Graphs Definition: an Directed Graph (Or Digraph) G = (V, E) Consists of a Nonempty Set V of Vertices (Or Nodes) and a Set E of Directed Edges (Or Arcs)
Chapter 10 Chapter Summary Graphs and Graph Models Graph Terminology and Special Types of Graphs Representing Graphs and Graph Isomorphism Connectivity Euler and Hamiltonian Graphs Shortest-Path Problems (not currently included in overheads) Planar Graphs (not currently included in overheads) Graph Coloring (not currently included in overheads) Section 10.1 Section Summary Introduction to Graphs Graph Taxonomy Graph Models Graphs Definition: A graph G = (V, E) consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints. Example: a b This is a graph with four vertices and five edges. d c Remarks: The graphs we study here are unrelated to graphs of functions studied in Chapter 2. We have a lot of freedom when we draw a picture of a graph. All that matters is the connections made by the edges, not the particular geometry depicted. For example, the lengths of edges, whether edges cross, how vertices are depicted, and so on, do not matter A graph with an infinite vertex set is called an infinite graph. A graph with a finite vertex set is called a finite graph. We (following the text) restrict our attention to finite graphs. Some Terminology In a simple graph each edge connects two different vertices and no two edges connect the same pair of vertices. Multigraphs may have multiple edges connecting the same two vertices. When m different edges connect the vertices u and v, we say that {u,v} is an edge of multiplicity m.
[Show full text]
A Brief Introduction to Spectral Graph Theory
A BRIEF INTRODUCTION TO SPECTRAL GRAPH THEORY CATHERINE BABECKI, KEVIN LIU, AND OMID SADEGHI MATH 563, SPRING 2020 Abstract. There are several matrices that can be associated to a graph. Spectral graph theory is the study of the spectrum, or set of eigenvalues, of these matrices and its relation to properties of the graph. We introduce the primary matrices associated with graphs, and discuss some interesting questions that spectral graph theory can answer. We also discuss a few applications. 1. Introduction and Definitions This work is primarily based on [1]. We expect the reader is familiar with some basic graph theory and linear algebra. We begin with some preliminary definitions. Definition 1. Let Γ be a graph without multiple edges. The adjacency matrix of Γ is the matrix A indexed by V (Γ), where Axy = 1 when there is an edge from x to y, and Axy = 0 otherwise. This can be generalized to multigraphs, where Axy becomes the number of edges from x to y. Definition 2. Let Γ be an undirected graph without loops. The incidence matrix of Γ is the matrix M, with rows indexed by vertices and columns indexed by edges, where Mxe = 1 whenever vertex x is an endpoint of edge e. For a directed graph without loss, the directed incidence matrix N is defined by Nxe = −1; 1; 0 corresponding to when x is the head of e, tail of e, or not on e. Definition 3. Let Γ be an undirected graph without loops. The Laplace matrix of Γ is the matrix L indexed by V (G) with zero row sums, where Lxy = −Axy for x 6= y.
[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]
Graph Homomorphisms with Complex Values: a Dichotomy Theorem ∗
SIAM J. COMPUT. c 2013 Society for Industrial and Applied Mathematics Vol. 42, No. 3, pp. 924–1029 GRAPH HOMOMORPHISMS WITH COMPLEX VALUES: A DICHOTOMY THEOREM ∗ † ‡ § JIN-YI CAI ,XICHEN, AND PINYAN LU Abstract. Each symmetric matrix A over C defines a graph homomorphism function ZA(·)on undirected graphs. The function ZA(·) is also called the partition function from statistical physics, and can encode many interesting graph properties, including counting vertex covers and k-colorings. We study the computational complexity of ZA(·) for arbitrary symmetric matrices A with algebraic complex values. Building on work by Dyer and Greenhill [Random Structures and Algorithms,17 (2000), pp. 260–289], Bulatov and Grohe [Theoretical Computer Science, 348 (2005), pp. 148–186], and especially the recent beautiful work by Goldberg et al. [SIAM J. Comput., 39 (2010), pp. 3336– 3402], we prove a complete dichotomy theorem for this problem. We show that ZA(·)iseither computable in polynomial-time or #P-hard, depending explicitly on the matrix A. We further prove that the tractability criterion on A is polynomial-time decidable. Key words. computational complexity, counting complexity, graph homomorphisms, partition functions AMS subject classifications. 68Q17, 68Q25, 68R05, 68R10, 05C31 DOI. 10.1137/110840194 1. Introduction. Graph homomorphism has been studied intensely over the years [28, 23, 13, 18, 4, 12, 21]. Given two graphs G and H, a graph homomorphism from G to H is a map f from the vertex set V (G)to V (H) such that, whenever (u, v)isanedgeinG,(f(u),f(v)) is an edge in H. The counting problem for graph homomorphism is to compute the number of homomorphisms from G to H.Forafixed graph H, this problem is also known as the #H-coloring problem.
[Show full text]
Multilayer Networks
Journal of Complex Networks (2014) 2, 203–271 doi:10.1093/comnet/cnu016 Advance Access publication on 14 July 2014 Multilayer networks Mikko Kivelä Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK Alex Arenas Departament d’Enginyeria Informática i Matemátiques, Universitat Rovira I Virgili, 43007 Tarragona, Spain Marc Barthelemy Downloaded from Institut de Physique Théorique, CEA, CNRS-URA 2306, F-91191, Gif-sur-Yvette, France and Centre d’Analyse et de Mathématiques Sociales, EHESS, 190-198 avenue de France, 75244 Paris, France James P. Gleeson MACSI, Department of Mathematics & Statistics, University of Limerick, Limerick, Ireland http://comnet.oxfordjournals.org/ Yamir Moreno Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50018, Spain and Department of Theoretical Physics, University of Zaragoza, Zaragoza 50009, Spain and Mason A. Porter† Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, by guest on August 21, 2014 Oxford OX2 6GG, UK and CABDyN Complexity Centre, University of Oxford, Oxford OX1 1HP, UK †Corresponding author. Email: [email protected] Edited by: Ernesto Estrada [Received on 16 October 2013; accepted on 23 April 2014] In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such ‘multilayer’ features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize ‘traditional’ network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion.
[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]
Incidence Matrices and Interval Graphs
Pacific Journal of Mathematics INCIDENCE MATRICES AND INTERVAL GRAPHS DELBERT RAY FULKERSON AND OLIVER GROSS Vol. 15, No. 3 November 1965 PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 3, 1965 INCIDENCE MATRICES AND INTERVAL GRAPHS D. R. FULKERSON AND 0. A. GROSS According to present genetic theory, the fine structure of genes consists of linearly ordered elements. A mutant gene is obtained by alteration of some connected portion of this structure. By examining data obtained from suitable experi- ments, it can be determined whether or not the blemished portions of two mutant genes intersect or not, and thus inter- section data for a large number of mutants can be represented as an undirected graph. If this graph is an "interval graph," then the observed data is consistent with a linear model of the gene. The problem of determining when a graph is an interval graph is a special case of the following problem concerning (0, l)-matrices: When can the rows of such a matrix be per- muted so as to make the l's in each column appear consecu- tively? A complete theory is obtained for this latter problem, culminating in a decomposition theorem which leads to a rapid algorithm for deciding the question, and for constructing the desired permutation when one exists. Let A — (dij) be an m by n matrix whose entries ai3 are all either 0 or 1. The matrix A may be regarded as the incidence matrix of elements el9 e2, , em vs. sets Sl9 S2, , Sn; that is, ai3 = 0 or 1 ac- cording as et is not or is a member of S3 .
[Show full text]
Arxiv:1602.02002V1 [Math.CO] 5 Feb 2016 Dept
a The Structure of W4-Immersion-Free Graphs Rémy Belmonteb;c Archontia Giannopouloud;e Daniel Lokshtanovf ;g Dimitrios M. Thilikosh;i ;j Abstract We study the structure of graphs that do not contain the wheel on 5 vertices W4 as an immersion, and show that these graphs can be constructed via 1, 2, and 3-edge-sums from subcubic graphs and graphs of bounded treewidth. Keywords: Immersion Relation, Wheel, Treewidth, Edge-sums, Structural Theorems 1 Introduction A recurrent theme in structural graph theory is the study of specific properties that arise in graphs when excluding a fixed pattern. The notion of appearing as a pattern gives rise to various graph containment relations. Maybe the most famous example is the minor relation that has been widely studied, in particular since the fundamental results of Kuratowski and Wagner who proved that planar graphs are exactly those graphs that contain neither K5 nor K3;3 as a (topological) minor. A graph G contains a graph H as a topological minor if H can be obtained from G by a sequence of vertex deletions, edge deletions and replacing internally vertex-disjoint paths by single edges. Wagner also described the structure of the graphs that exclude K5 as a minor: he proved that K5- minor-free graphs can be constructed by \gluing" together (using so-called clique-sums) planar graphs and a specific graph on 8 vertices, called Wagner's graph. aEmails of authors: [email protected] [email protected], [email protected], [email protected] b arXiv:1602.02002v1 [math.CO] 5 Feb 2016 Dept.
[Show full text]
Multidimensional Network Analysis
Universita` degli Studi di Pisa Dipartimento di Informatica Dottorato di Ricerca in Informatica Ph.D. Thesis Multidimensional Network Analysis Michele Coscia Supervisor Supervisor Fosca Giannotti Dino Pedreschi May 9, 2012 Abstract This thesis is focused on the study of multidimensional networks. A multidimensional network is a network in which among the nodes there may be multiple different qualitative and quantitative relations. Traditionally, complex network analysis has focused on networks with only one kind of relation. Even with this constraint, monodimensional networks posed many analytic challenges, being representations of ubiquitous complex systems in nature. However, it is a matter of common experience that the constraint of considering only one single relation at a time limits the set of real world phenomena that can be represented with complex networks. When multiple different relations act at the same time, traditional complex network analysis cannot provide suitable an- alytic tools. To provide the suitable tools for this scenario is exactly the aim of this thesis: the creation and study of a Multidimensional Network Analysis, to extend the toolbox of complex network analysis and grasp the complexity of real world phenomena. The urgency and need for a multidimensional network analysis is here presented, along with an empirical proof of the ubiquity of this multifaceted reality in different complex networks, and some related works that in the last two years were proposed in this novel setting, yet to be systematically defined. Then, we tackle the foundations of the multidimensional setting at different levels, both by looking at the basic exten- sions of the known model and by developing novel algorithms and frameworks for well-understood and useful problems, such as community discovery (our main case study), temporal analysis, link prediction and more.
[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]