Based On The Research Of Topological Index Of The Tree Of The Distance

Graph theory and its application is a important branch of combinatorial mathematics,it has many important applications in mathematics and other subject-s.In recent years graph theory is combined with other subjects and the formation of distinctive multiple branches.For example,Algebraic graph theory,topological graph theory,chemical graph theory,random graph,graph algorithm.The topologi-cal index of graph in physics,chemistry,information network,science has important applications and is one of the most active research fields in graph theory.Since H.Wiener raised the first molecular topological index,Wiener index in 1975,hundreds of molecular topological index has been widely researched and application.The connective eccentricity index and the total eccentricity are based on the topological index of graph of the distance.This paper mainly studies the graph on the operation of splitting a vertex for the influence of the connective eccentricity index and the total eccentricity.This paper is divided into four parts and the main structure of the article is as follows.In chapter 1:we introduce the fundamental concepts and properties of graphs and molecular topological index,also the domestic and foreign researching situations.In chapter 2:In this chapter,we study the simple connected graph on the operation of splitting a vertex for the influence of the connective eccentricity index and the total eccentricity,The extremal trees of the connective eccentricity index for the trees with a given degree sequence.We prove that the upper bound and the lower bound of the connective eccentricity index of an n-vertex tree with r branching vertices.In chapter 3:we study the upper bound and the lower bound of the total eccentricity of trees with given number of branching vertices.In chapter 4:We prove that the lower bound of the Wiener index of an n-vertex tree with r(1?r?n/2-1)branching vertices.