Russell Merris

Laplacian matrices of graphs: a survey

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 Equivalence, Chemical Applications, and Immanants.

The Laplacian spectrum of a graph

Let G be a graph. The Laplacian matrix

The Laplacian Spectrum of a Graph II

L(G) = D(G) - A(G)

Laplacian graph eigenvectors

is the difference of the diagonal matrix of vertex degrees and the 0-1 adjacency matrix. Various aspects of the spectrum of

A survey of graph laplacians

L(G)

A note on Laplacian graph eigenvalues

are investigated. Particular attention is given to multiplicities of integer eigenvalues and to the effect on the spectrum of various modifications of G.

Degree maximal graphs are Laplacian integral

Let G be a graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then L(G) = D(G) - A(G) is the Laplacian matrix of G. The first section of this paper is devoted to properties of Laplacian integral graphs, those for which the Laplacian spectrum consists entirely of integers. The second section relates the degree sequence and the Laplacian spectrum through majorization. The third section introduces the notion of a d-cluster, using it to bound the multiplicity of d in the spectrum of L(G).

The distance spectrum of a tree

Abstract If G is a graph, its Laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix. The main thrust of the present article is to prove several Laplacian eigenvector “principles” which in certain cases can be used to deduce the effect on the spectrum of contracting, adding or deleting edges and/or of coalescing vertices. One application is the construction of two isospectral graphs on 11 vertices having different degree sequences, only one of which is bipartite, and only one of which is decomposable.

Permanental polynomials of graphs

Let G be a graph on n vertices. Its Laplacian is the n-by-n matrix L(G)−D(G)−A(G), where D(G) is the diagonal matrix of vertex degrees and A(G) is the (0,1)-adjacency matrix of G. This article surveys recent results on graph Laplacians.

Ordering trees by algebraic connectivity

Abstract Let G = (V,E) be a graph on n vertices. Denote by d(v) the degree of v ∈ V and by m(v) the average of the degrees of the vertices of G adjacent to v. Then b(G) = max{m(v) + d(v): v ∈ V} is an upper bound for the Laplacian spectral radius of G; hence, n − b(GC) is a lower bound for the algebraic connectivity of G in terms of the vertex degrees of its complement.