Spectral graph theory

Abstract

With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian and the adjacency matrix of a graph. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properies (the eigenvalues and eigenvectors) of these matrices provide useful information about the structure of the graph. It turns out that for regular graphs, the information one can deduce from one matrix representation (e.g., the adjacency matrix) is similar to the information one can deduce from other representations (such as the Laplacian). We remark that for nonregular graphs, this is not the case, and the choice of matrix representation may make a significant difference. We shall not elaborate on this issue further, as our main concern here will be either with regular or nearly regular graph. The adjacency matrix of a connected undirected graph is nonnegative, symmetric and irreducible (namely, it cannot be decomposed into two diagonal blocks and two off-diagonal blocks, one of which is all-0). As such, standard results n linear algebra, including the Perron-Frobenius theorem, imply that: