Generalized spectral characterizations of almost controllable graphs

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