Smith Normal Form and the generalized spectral characterization of graphs

Abstract

Spectral characterizations of graphs are an important topic in spectral graph theory, which has received a lot of attention in recent years. It is generally very hard to show that a given graph is determined by its spectrum. Recently, Wang [9] gave a simple arithmetic condition for graphs being determined by their generalized spectra. Let G be a graph on n vertices with adjacency matrix A, and W=[e,Ae,…,An−1e] (e is the all-one vector) be the walk-matrix of G. A theorem of Wang [9] states that if 2−⌊n/2⌋det⁡W (which is always an integer) is odd and square-free, then G is determined by the generalized spectrum. In this paper, we find a new and short route which leads to a stronger version of the above theorem. The result is achieved by using the Smith Normal Form of the walk-matrix of G. The proposed method gives a new insight in dealing with the problem of generalized spectral characterization of graphs.