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

A graph GG is said to be determined by its spectrum   (DS for short), if any graph having the same spectrum as GG is necessarily isomorphic to GG. 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.