Generalized spectral characterizations of almost controllable graphs

Abstract

Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph is determined by its spectrum. In Wang (2017), the author gave a simple arithmetic condition for a family of graphs being determined by their generalized spectra. However, the method applies only to a family of the so called controllable graphs; it fails when the graphs are non-controllable.In this paper, we introduce a class of non-controllable graphs, called almost controllable graphs, and prove that, for any pair of almost controllable graphs G and H that are generalized cospectral, there exist exactly two rational orthogonal matrices Q with constant row sums such that QTA(G)Q=A(H), where A(G) and A(H) are the adjacency matrices of G and H, respectively. The main ingredient of the proof is the Binet–Cauchy formula. As an application, we obtain a simple criterion for an almost controllable graph G to be determined by its generalized spectrum, which in some sense extends the corresponding result for controllable graphs.