An arithmetic criterion for graphs being determined by their generalized Aα-spectra

Abstract

Let G be a graph on n vertices, its adjacency matrix and degree diagonal matrix are denoted by A(G) and D(G), respectively. In 2017, Nikiforov [20] introduced the matrix Aα(G)=αD(G)+(1−α)A(G) for α∈[0,1]. The Aα-spectrum of a graph G consists of all the eigenvalues (including the multiplicities) of Aα(G). A graph G is said to be determined by the generalized Aα-spectrum (or, DGAαS for short) if whenever H is a graph such that H and G share the same Aα-spectrum and so do their complements, then H is isomorphic to G. In this paper, when α is rational, we present a simple arithmetic condition for a graph being DGAαS. More precisely, put Acα:=cαAα(G), here cα is the smallest positive integer such that Acα is an integral matrix. Let W˜α(G)=[1,Acα1cα,…,Acαn−11cα], where 1 denotes the all-ones column vector. We prove that if det⁡W˜α(G)2⌊n2⌋ is an odd and square-free integer and the rank of W˜α(G) is full over Fp for each odd prime divisor p of cα, then G is DGAαS except for even n and odd cα(⩾3). By our obtained results in this paper we may deduce the main results in [24] and [27].