The Spectral Radius Of Trees

The spectral of graphs is an important direction of investigations in algebraic graph theory, and it mainly studies the adjacency spectrum and the Laplacian spectrum of graphs. This direction associates the graph in graph theory with the matrix in algebra by the matrix representation of a graph, then studies the algebraic properties of graphs by the methods in graph theory and algebra.In this paper we study the spectral radius of trees, which is a special and important direction in the theory of spectral of graphs. There is now an extensive literature that investigates the spectral radius of trees. On the basis of previous results, this article will further investigate the relationships between the spectral radius and variables of trees such as diameter, independence number, matching number, etc. The main content can be divided into three chapters.In the first chapter, we give an overview of spectral of graphs, introduce some related definitions and notations, and explain the structure of this paper.In the second chapter, we first study some properties of adjacent spectral radius of trees with fixed diameter and weight set, then determine the weighted tree with the largest adjacent spectral radius, and at last study the relationships between the adjacent spectral radius of weighted trees and the independence number, matching number, covering number, etc.In the third chapter, we investigate the Laplacian spectral radius of caterpillar trees with fixed diameter and determine the caterpillar tree with the largest Laplacian spectral radius. Furthermore, several properties of caterpillar tree are discussed.