In this chapter we discuss several instances of the following problem:

Given the spectrum, or some spectral characteristics of a graph, determine all graphs from a given class of graphs having the given spectrum, or the given spectral characteristics.

In some cases, the solution of such a problem can provide a characterization of a graph up to isomorphism (see Section 4.1). In other cases we can deduce structural details (see also Chapter 3). Non-isomorphic graphs with the same spectrum can arise as sporadic exceptions to characterization theorems or from general constructions. Accordingly, Section 4.2 is devoted to cospectral graphs; we include comments on their relation to the graph isomorphism problem, together with various examples and statistics. We also discuss the use of other graph invariants to strengthen distinguishing properties. In particular, in Section 4.3, we consider characterizations of graphs by eigenvalues and angles.

Spectral characterizations of certain classes of graphs

In this section we investigate graphs that are determined by their spectra. The three subsections are devoted to (i) elementary characterizations, (ii) characterizations of graphs with least eigenvalue -2, and (iii) characterizations of special types. In the case of (i), a graph is uniquely reconstructed from its spectrum, while in cases (ii) and (iii) various exceptions occur due to the existence of cospectral graphs.