The Distance (Signless) Laplacian Spectral Radius Of Some Graphs

In the development of graph theory,many matrices have been introduced into the research of graph theory,such as incidence matrix,adjacency matrix,Laplacian matrix,signless laplacian matrix and distance matrix etc.By studying the eigenvalues of these matrices,people can describe the structure of the graph.In 2013,inspired by Laplacian matrix and signless Laplacian matrix,Aouchiche and Hansen introduced distance Laplacian matrix and distance signless Laplacian matrix of a connected graph and studied their spectra.Let T r(G)be the diagonal matrix of vertex transmissions of G and D(G)be the distance matrix of G.Then L(G)= T r(G)-D(G)and (G)= T r(G)+ D(G)are the distance Laplacian matrix and distance signless Laplacian matrix of G,respectively.The largest eigenvalues of L(G)and (G)are called the distance Laplacian spectral radius and distance signless Laplacian spectral radius of G,respectively.In this thesis,we study the distance Laplacian spectral radius and distance signless Laplacian spectral radius of bicyclic graphs,and the Hamiltonian connectivity and traceability of the graph.This thesis is organized as follows.Chapter 1 is the introduction.We introduce the background of the graph theory,and basic definitions and notations related to this thesis,the known results of the distance Laplacian spectral radius and distance signless Laplacian spectral radius of graphs and the main work of this thesis.In Chapter 2,we present some useful lemmas which are related to the distance Laplacian matrix and distance Laplacian spectral radius of graphs,and prove two results on the unit eigenvectors corresponding to the distance Laplacian spectral radius of two special graphs.By comparing the distance Laplacian spectral radii of some special graphs,we determine the unique graphs with the maximum and minimum distance Laplacian spectral radius in the class of all bicyclic graphs,respectively.Moreover,we give two sufficient conditions on distance Laplacian spectral radius for a graph to be Hamiltonian connected and traceable from every vertex,respectively.Furthermore,we obtain a sufficient condition for a graph to be traceable in terms of the distance Laplacian spectral radius of the complement of a graph G.In Chapter 3,we present some useful lemmas which are related to the distance signless Laplacian matrix and distance signless Laplacian spectral radius of graphs.By comparing the distance signless Laplacian spectral radii of some special graphs,we determine the unique graph with the maximum distance signless Laplacian spectral radius among all bicyclic graphs.In Chapter 4,we summary the thesis and give further research directions.