Laplacian Matrices of Graphs: a Survey
Laplacian Matrices of Graphs: A Survey Russell Merris* Department of Mathematics and Computer Science California State University Hayward, Cal$ornia 94542 Dedicated to Miroslav Fiedler in commemoration of his retirement. Submitted by Jose A. Dias da Silva ABSTRACT Let G be a graph on n vertices. Its Laplacian matrix is the n-by-n matrix L(G) = D(G) - A(G), where A(G) is the familiar (0, 1) adjacency matrix, and D(G) is the diagonal matrix of vertex degrees. This is primarily an expository article surveying some of the many results known for Laplacian matrices. Its six sections are: Introduction, The Spectrum, The Algebraic Connectivity, Congruence and Equiva- lence, Chemical Applications, and Immanants. 1. INTRODUCTION Let G = (V, E) be a graph with vertex set V = V(G) = {u,, 02,. , wn} and edge set E = E(G) = {el, e2,. , e,). For each edge ej = {vi, ok), choose one of ui, vk to be the positive “end’ of ej and the other to be the negative “end.” Thus G is given an orientation [ll]. The vertex-edge inci- dence matrix (or “cross-linking matrix” [33]) afforded by an orientation of G is the n-by-m matrix Q = Q(G) = (qi .I, where qij = + 1 if q is the positive end of ej, - 1 if it is the negative en d , and 0 otherwise. *This article was prepared in conjunction with the author’s lecture at the 1992 conference of the International Linear Algebra Society in Lisbon. Its preparation was supported by the National Security Agency under Grant MDA904-90-H-4024.
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