Generalized spectral characterization of rooted product graphs

Abstract

A graph G is said to be determined by its generalized spectrum (DGS for short) if for any graph H, whenever H and G are cospectral with cospectral complements, then H is isomorphic to G. In Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112–127.], the authors gave an explicit method of constructing large DGS graphs from small ones. More precisely, let G be a graph on n vertices with adjacency matrix A, and (e denotes the all-one vector) be its walk-matrix. The authors proved, among others, that the rooted product graph (i.e. a graph obtained from G by adding a pendent edge at every vertex of G) is DGS whenever (n is even) and the constant term of the characteristic polynomial of graph G is . In this paper, we shall extend the above result in two directions, namely, in the above-rooted graph , we may replace the pendent edge with a path for k = 3 or 4; we are also able to enlarge the family of graphs G satisfying to a family of graphs , in which every graph G satisfies that is odd and square-free. We prove that, under certain conditions, the new rooted product graphs are still DGS. In particular, based on the above result we obtain infinite sequences of DGS graphs for some DGS graph G.