On The Extremal Wiener Polarity Index Of Unicyclic Graphs With Given Diameter D

Let G= (V, E) be a connected simple graph. The distance between two vertices u and v in G, denoted by dc(u, v), is the length of a shortest path between u and v in G. The Wiener polarity index of a graph G, denoted by WP(G), is the number of unordered pairs of vertices{u, v} of G such that dG(u, v) = 3,i.e., WP(G):=|{{u,v}|d(u,v)= 3,u,v ∈ V(G)}|.The name "Wiener polarity index" was introduced by Harold Wiener, a famous mathematics chemist and theoretical chemist, in 1947, when he was s-tudying the acyclic molecules [32]. In the same paper, Wiener also introduced another index for acyclic molecules, called Wiener index and defined by W(G):= ∑{u,v}(?)V dG(u,v).Wiener [32] applied a linear formula of W and Wp to calculate the boiling points tB of the paraffins, i.e., tB= aW + bWp + c, where a, b and c are constants for a given isomeric group. Soon after those two indices were proposed, the Wiener index W(G) became popular and studied by many researchers. In fact, early before the concept of Wiener index was put forward, the theorists on graph theory had already begun the research on the average distance related to this index. However, the Wiener polarity index cause the researchers’ attention just in recent years and only a few results have been obtained. In addition, the Wiener polarity index has close relationships with the counting problem of the paths in a graph, which implies that the study for the Wiener polarity index has great significance in theory and applications.The Wiener index and the Wiener polarity index are both topological indices. One of the fundamental problems in the study of the Wiener polarity index is which graph has the maximal or minimal Wiener polarity index within a given graph class. This thesis is devoted to determine the extremal graphs, i.e., the extremal Wiener polarity index of unicyclic graphs with a given diameter.In Chapter 1, we first introduce the background of the Wiener polarity index, then give the basic notations and terminologies used in this thesis. At last, we list an overview of the main results of this thesis.The second chapter is devoted to giving a result about the Wiener polarity index of trees and unicyclic graphs.In Chapter 3, we introduce a new method which applies some operations on unicyclic graphs to insure that the Wiener polarity index will not increase after those operations. After that, we give the minimum Wiener polarity index and also characterize the extremal graphs among all unicyclic graphs with order n and diameter d.In the last chapter, some different operations are proposed, which also insure that the Wiener polarity index will not decrease after any one of them. Finally, we will characterize the extremal graphs with respect to the maximum Wiener polarity index among all unicyclic graphs with order n and diameter d.