The spectrum of a graph has been widely used in graph theory to characterise the properties of a graph and extract information from its structure. It has been less popular as a representation for pattern matching for two reasons. Firstly, more than one graph may share the same spectrum. It is well known, for example, that very few trees can be uniquely specified by their spectrum. Secondly, the spectrum may change dramatically with a small change structure. In this paper we investigate the extent to which these factors affect graph spectra in practice, and whether they can be mitigated by choosing a particular matrix representation of the graph. There are a wide variety of graph matrix representations from which the spectrum can be extracted. In this paper we analyse the adjacency matrix, combinatorial Laplacian, normalised Laplacian and unsigned Laplacian. We also study the use of the spectrum derived from the heat kernel matrix and path length distribution matrix. We investigate the cospectrality of these matrices over large graph sets and show that the Euclidean distance between spectra tracks the edit distance over a wide range of edit costs, and we analyse the stability of this relationship. We then use the spectra to match and classify the graphs and demonstrate the effect of the graph matrix formulation on error rates.