1 - A background in graph spectra

Summary

In Section 1.1 we introduce notation and terminology which will be used throughout the book. The limitations of the spectrum as a graph invariant are illustrated by the discussion of non-isomorphic cospectral graphs in Section 1.2. In Section 1.3 we describe the extent to which certain classes of graphs are characterized by spectral properties, and in Section 1.4 we discuss ways of extending the spectrum to a set of invariants which together are sufficient to characterize a graph.

Basic notions and results

A comprehensive treatment of the theory of graph spectra is given in the monograph [CvDS], while some of the underlying results from matrix theory are given in Appendix A. Here we present only those basic notions and further results which are needed frequently in other chapters. We recommend as general references the texts by Biggs [Big] and Harary [Har2].

The adjacency matrix of a (multi)(di)graph G, with vertex set {1, 2, …, n}, is the n × n matrix A = (a_ij) whose (i, j)-entry a_ij is equal to the number of edges, or arcs, originating at the vertex i and terminating at the vertex j. Two vertices of G are said to be adjacent if they are connected by an edge or arc. Unless we indicate otherwise we shall assume that G is an undirected graph without loops or multiple edges.