Aα-Spectral Characterizations of Some Joins

Tingzeng Wu,Tian Zhou

doi:10.1155/2020/8294312

Abstract

LetGbe a graph withnvertices. For every realα∈0,1, writeAαGfor the matrixAαG=αDG+1−αAG, whereAGandDGdenote the adjacency matrix and the degree matrix ofG, respectively. The collection of eigenvalues ofAαGtogether with multiplicities are called theAα-spectrum ofG. A graphGis said to be determined by itsAα-spectrum if all graphs having the sameAα-spectrum asGare isomorphic toG. In this paper, we show that some joins are determined by theirAα-spectra forα∈0,1/2 or 1/2,1.

Highlights

I 1 is called Aα-characteristic polynomial, where I is the identity matrix of order n. e theory of Aα-characteristic polynomial of a graph is well elaborated [1,2,3,4,5,6,7,8]
It is interesting to characterize which graph is determined by some graph spectrum [9,10,11]. e problem was raised by Gunthard and Primas [12] in 1956 with motivations from chemistry
We focus on Problem 1 above, and we prove that some join graphs are Aα-DS graphs

Summary

Introduction

I 1 is called Aα-characteristic polynomial, where I is the identity matrix of order n. e theory of Aα-characteristic polynomial of a graph is well elaborated [1,2,3,4,5,6,7,8]. A graph is called an Aα-DS graph if it is determined by its Aα-spectrum, meaning that there exists no other graph that is nonisomorphic to it but Aα-cospectral with it. It is interesting to characterize which graph is determined by some graph spectrum [9,10,11]. Lin et al [15] considered the problem which graph is determined by its Aα-spectrum? They gave some characterizing properties of Aα-spectrum and proposed the following problem. Liu and Lu [16] discussed the problem which join graph is determined by its Q-spectrum?

Preliminaries

Main Results

G48 G51 G57 G522 G517 G523 G525