The Multiplicity Of The Distance Eigenvalues Of Graphs

The distribution theory of the eigenvalues and the limit point theory of eigenvalues of a graph are important subjects in algebraic graph theory.This paper mainly investigates the distribution of the eigenvalues of the distance matrix of a connected graph.Let D（G）=(d_ij)be the distance matrix of the connected graph G,whered d_ijdenotes the distance between vertices v_i and v_j.One can define the two Laplacian matrices of the distance matrix,namely,the distance Laplacian matrix and the distance signless Laplacian matrix.The adjacency matrix,the Laplacian matrix,the signless Laplacian matrix,the distance matrix,the distance Laplacian matrix and the distance signless Laplacian matrix are six important graph matrices.Their algebraic properties（spectral properties）can reflect the good structural properties of graphs.In the known literatures,researchers at home and abroad in the field of spectral graph theory focus on the characterization problem of graphs with large multiplicity of the diatance spectral radius,the distance Laplacian spectral radius and the distance signless Laplacian spectrum radius,respectively.Furthermore,similar questions are also studied on the least distance eigenvalues and the least distance signless Laplacian eigenvalue.Inspired by the above results,we focus on the characterization of the set of graphs,where the multiplicity of some distance eigenvalue is equal to n-2.It is worth to noting that the biggest difference between the research results in this paper and those in the literature is that this paper does not specify which the distance eigenvalue of a graph.Therefore,the research questions in this paper contain and extend those in the literature,which have extensive and theoretical significance.In the first chapter,we firstly introduce the history and background of spectral graph theory,and literatures related to the research questions.Secondly,we define some impor-tant concepts and symbols used in the paper.Finally,we give the main results of this paper briefly.The second chapter is the core of the thesis.First,the relevant research results in the literature are introduced.Then we turn to the proof of the main results.By a series of assertions and lemmas,we characterize the connected graphs,where the multiplicity of some distance eigenvalue is n-2 in this paper.