Graphs determined by their $A_α$-spectra

Abstract

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. Definefor any real α ∈ [0, 1]. The collection of eigenvalues of Aα(G) together with multiplicities are called the Aα-spectrum of G. A graph G is said to be determined by its Aα-spectrum if all graphs having the same Aα-spectrum as G are isomorphic to G. We first prove that some graphs are determined by its Aα-spectrum for 0 ≤ α < 1, including the complete graph Km, the star K1,n−1, the path Pn, the union of cycles and the complement of the union of cycles, the union of K2 and K1 and the complement of the union of K2 and K1, and the complement of Pn. Setting α = 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 Aα-spectrum if and only if the join G ∨ Km is determined by its Aα-spectrum for 1 2 < α < 1. Furthermore, we also show that the join Km ∨ Pn is determined by its Aα-spectrum for 1 2 < α < 1. In the end, we pose some related open problems for future study.