Paper Info
Title
An Arithmetic Criterion For Graphs Being Determined By Their Generalized A(Alpha)-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 A(alpha)(G) = alpha D(G) +(1 - alpha) A(G) for alpha is an element of[0, 1]. The A(alpha)-spectrum of a graph G consists of all the eigenvalues (including the multiplicities) of A(alpha)(G). A graph G is said to be determined by the generalized A(alpha)-spectrum (or, DGA(alpha)S for short) if whenever His a graph such that Hand G share the same A(alpha)-spectrum and so do their complements, then His isomorphic to G. In this paper, when alpha is rational, we present a simple arithmetic condition for a graph being DGAaS. More precisely, put A(c alpha) := c(alpha)A(alpha)(G), here c(alpha) is the smallest positive integer such that A(c alpha) is an integral matrix. Let (W) over tilde (alpha)(G) = [1, A(c alpha)1/c(alpha), ..., A(c alpha)(n-1)1/c(alpha)], where 1 denotes the all-ones column vector. We prove that if det (W) over tilde (alpha)(G)/2[n/2] is an odd and square-free integer and the rank of (W) over tilde (alpha)(G) is full over F-p for each odd prime divisor p of c(alpha), then Gis DGA(alpha)S except for even nand odd c(alpha)(>= 3). By our obtained results in this paper we may deduce the main results in [24] and [27]. (C) 2021 Elsevier B.V. All rights reserved.
Year
DOI
Venue
2021
10.1016/j.disc.2021.112469
DISCRETE MATHEMATICS
Keywords
DocType
Volume
Generalized A(alpha)-spectrum, Cospectral, A(alpha)-matrix, Walk matrix
Journal
344
Issue
ISSN
Citations 
8
0012-365X
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Shuchao Li118335.15
Wanting Sun201.35