2021/03/06 by Li, Shuchao, Sun, Wanting
#05C50 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2103.04010
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 \cite0007 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 Wα(G)=[\bf 1,\fracAcα\bf 1cα,…, \fracAcαn-1\bf 1cα], where \bf 1 denotes the all-ones vector. We prove that if \fracdet Wα(G)2\lfloor(n)/(2)\rfloor is an odd and square-free integer and the rank of Wα(G) is full over \mathbbFp for each odd prime divisor p of cα, then G is DGAαS except for even n and odd cα (\geqslant 3). By our obtained results in this paper we may deduce the main results in \cite0005 and \cite0002.