The Characterization Of Graphs With A Large Distance Eigenvalue Multiplicity

Spectral graph theory is a very important area in algebraic graph theory and combi-natorial matrix theory.It mainly investigates the properties of eigenvalues and eigenvectors of graph matrices.Graph matrices mainly include the adjacency matrix,the Laplacian matrix,the signless Laplacian matrix,the distance matrix,the distance Laplacian matrix and the distance signless Laplacian matrixRecently,many experts and scholars investigate the graphs with a large distance eigenvalue multiplicity.They characterized the graphs which have a distance eigenvalue with multiplicity n-2,and the graphs with exactly two distance eigenvalues different from-1 and-3.There are also many results on the distance Laplacian eigenvalues and the distance signless Laplacian eigenvalues.They characterized the graphs which have a distance Laplacian eigenvalue with multiplicity at least n-3,and the graphs which have a distance signless Laplacian eigenvalue with multiplicity at least n-2.This paper focuses on the distance eigenvalues of graphs.Let G be a simple and undirected connected graph.The distance,denoted by d(u,v)or duv,is the length of the shortest path between u and v.The distance matrix of G is an n × n matrix denoted by D(G):=(dij).We will study the graphs with a large distance eigenvalue multiplicity in this paperThe main results are as followsIn the first chapter,we introduce the research background and progress on the dis-tance matrix,and give many useful concepts and notation.Furthermore,we present the motivation of the problems in this paperIn the second chapter,we present the simple method to characterize the graphs which have an eigenvalue with multiplicity n-2,and determine the graphs which have an eigenvalue with multiplicity n-3.In the third chapter,we determine the graphs with exactly four distance eigenvalues different from-1.