On The Distribution Of Laplacian Eigenvalues Of Trees

The spectral of graph theory is an important branch of the research direction of alge-braic graph theory.According to the algebraic representation of the adjacency matrix,Laplacian matrix,incidence matrix and so on,The spectral of graph theory study the topological properties and its certainty of the graph by mainly using the combination matrix theory(especially the non negative matrix theory).Since the Laplacian matrix is the difference between the degree diagonal matrix D(G)and the adjacency matrix A(G),so it can react the structural properties of the graph more than the adjacency matrix,which makes the Laplacian spectrum more extensive,such as its application in the fields of quantum chemistry,computer science and so on.This paper mainly stud-ies the distribution of Laplacian eigenvalues of trees by the diagonalization algorithm,and then gives the Laplacian energy bound of a class of trees by the distribution of the Laplacian eigenvalues of the trees.The content distribution is as follows:In Chapter 1 we describe the background and significance of the main research directions,and summarizes the main contents of this paper.In Chapter 2 we introduce a diagonalization algorithm,which is the core tool to prove the conclusion of this paper.In Chapter 3 we introduce the research status of the Laplacian eigenvalues distri-bution of the trees.In chapter 4 we give the main results of this paper.We discuss the distribution of Laplacian eigenvalues of trees,specially Spider trees and trees with diameter 5,and prove the Laplacian energy of a special tree Jn,4(p,k,r)is greater than the path Pn with the same number of vertex.In Chapter 5 we give a brief summary of the research contents in this paper.