Some Research On Eccentricity Spectrum Of Graphs And Related Problems

The M-eigenvalues of a graph G are those of its graph matrices M（G）.The M-spectra of a graph consists of its M-eigenvalues.Then the spectral theory of graphs mainly studies the properties of graphs by studying the eigenvalues and eigenvectors of graph matrices.As we known,the graph matrices mainly include the adjacency matrix,the distance matrix,Laplacian matrix and so on.Let D（G）=(d_ij)be the distance matrix of G,where d_ij is the distance between the vertices i and j.Then the matrix ε（G）can be obtained from the distance matrix D（G）by retaining the largest distances in each row and each column and replacing the remaining entries with zeros.In this paper,we mainly studies the spectral problems of eccentricity matrix and its related distance matrices.The distance spectral radius of G is the largest D-eigenvalue.In Chapter 2,the lower bounds of the second、third and forth largest ε-eigenvalue of the graph are given,which are related to the diameter of this graph.In Chapter 3,the ε-spectrum of the center graph of the triangle-free regular graph is given,and the ε-spectra of the center point union graph and center edge union graph of two triangle-free regular graphs are also described.In chapter 4,we consider subset NC_n containing non-caterpillar tree and subset NS_n containing non-star tree,and characterize the graphs with maximum distance Laplacian radii in NC_n,NS_n and NC_n ∩ NS_n.In addition,three candidate graphs for obtaining the forth largest distance Laplacian spectral radius in all order n trees are determined.