The Normality And Sum Of Normalities Of Trees And Unicyclic Graphs Involving Diameter And Matching Number

Graph theory mainly studies the internal structure contained in graphs,in which the research on the polynomials of graphs is an important field.One may characterize the structure of graphs with the help of some important parameters.This is a hot topic in the study of graph theory.The normality of a vertex v is the minimum distance from v to all peripheral vertices in G.The sum of normalities of G is denoted by Norm(G).The difference between the eccentricity and normality at a vertex v is denoted by ?G(v).Then the sum of all the ?G(v)is denoted by ?(G).In this paper,we mainly study the normality of trees and unicyclic graphs with given diameters and matching numbers,and then obtain the polar graphs and some extreme values.The problem of ?(·)extremum for trees and unicyclic graphs is also studied.The concrete content is in the following:·In Chapter 1,we first introduce the background and significance of the research,including the development of a representative at home and abroad regarding this aspect;Then give some necessary notations,terminologies and the main conclusions of this paper.·In Chapter 2,by studying the extremal sum of normalities of n-vertex tree with given matching number,we determine the changes of tree normality in various graph transformations,and then describe the structure and extreme value of the pole graph when tree normality reaches its maximum value.·In Chapter 3,studying the extremal sum of normalities of n-vertex unicyclic graphs,and then describe the structure and extreme value of the pole graph when unicyclic graphs normality reaches its maximum value.·In Chapter 4,mainly studying the extremal problems on ?(·)of trees and unicyclic graphs.·In Chapter 5,we summarize the main results in this paper and give some prospects for further research in the future.