Abstract
Let G be an n-vertex graph with adjacency matrix A and let e be the all-one vector. Then the walk matrix of G is defined as W=[e,Ae,A2e,…,An−1e]. We call G controllable if the rank of W is n and noncontrollable otherwise. A graph G is determined by the generalized spectrum, DGS for short, if any graph H having the same generalized spectrum as G must be isomorphic to G. In Wang [13], the author proposed a method for a family of controllable graphs to be DGS. However, the method fails for noncontrollable graphs, and it turns out that proving a noncontrollable graph to be DGS is more difficult than proving a controllable graph to be DGS. Up to now, there are just a few attempts to investigate generalized spectral characterizations of noncontrollable graphs for which the rank of W is n−1. In this paper, we focus on spectral characterizations of a new family of noncontrollable graphs for which the rank of W is n−2, denoted by Gn. We present an arithmetic criterion to determine whether a given graph is DGS in Gn.
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