Graphs determined by their Aα-spectra

Let G be a graph with n vertices, and let A(G) and D(G) denote respectively the adjacency matrix and the degree matrix of G. Define A(alpha)(G) = alpha D(G) + (1 - alpha)A(G) for any real alpha is an element of [0, 11]. The collection of eigenvalues of A(alpha)(G) together with multiplicities are called the A(alpha)-spectrum of G. A graph G is said to be determined by its A(alpha)-spectrum if all graphs having the same A(alpha)-spectrum as G are isomorphic to G. We first prove that some graphs are determined by their A(alpha)-spectra for 0 <= alpha < 1, including the complete graph K-n, the union of cycles, the complement of the union of cycles, the union of copies of K2 and K-1, the complement of the union of copies of K-2 and K-1, the path Pei, and the complement of P-n. Setting alpha = 0 or 1/2, those graphs are determined by A- or Q-spectra. Secondly, when G is regular, we show that G is determined by its A(alpha)-spectrum if and only if the join G v Km (m >= 2) is determined by its A(alpha)-spectrum for 1/2 < alpha < 1. Furthermore, we also show that the join K-m boolean OR P-n (m, n >= 2) is determined by its A(alpha)-spectrum for 1/2 < alpha < 1. In the end, we pose some related open problems for future study. (C) 2018 Published by Elsevier B.V.