Characterization of Graphs with an Eigenvalue of Large Multiplicity

Linming Qi,Weiliang Zhao,Lu Liu,Lianying Miao

doi:10.1155/2020/3054672

Abstract

Let G be a simple and undirected graph. The eigenvalues of the adjacency matrix of G are called the eigenvalues of G. In this paper, we characterize all the n-vertex graphs with some eigenvalue of multiplicity n−2 and n−3, respectively. Moreover, as an application of the main result, we present a family of nonregular graphs with four distinct eigenvalues.

Highlights

All graphs here considered are simple, undirected, and connected
The rank of the adjacency matrix AðGÞ of G is called the rank of G, written as rðGÞ
An independent set of G is a subset of VðGÞ such that there is no edge between any two vertices

Summary

Introduction

All graphs here considered are simple, undirected, and connected. Let G be a graph with vertex set VðGÞ = fv1, v2, ⋯, vng. Let mðρiÞ be the multiplicity of an eigenvalue ρi of a graph G. Let G be a graph of order n with an eigenvalue ρ.

Results

Conclusion