A sufficient condition for a family of graphs being determined by their generalized spectra

A graph G is said to be determined by its spectrum (DS for short), if any graph having the same spectrum as G is necessarily isomorphic to G . One important topic in the theory of graph spectra is how to determine whether a graph is DS or not. The previous techniques used to prove a graph to be DS heavily rely on some special properties of the spectrum of the given graph. They cannot be applied to general graphs. In this paper, we propose a new method for determining whether a family of graphs (which have no special properties) are DS with respect to their generalized spectra. The method is obtained by employing some arithmetic properties of a certain matrix associated with a graph. Numerical examples are further given to illustrate the effectiveness of the proposed method.