Topological Indices Of Graphs And Their Applications

In theoretical chemistry,chemists have found that there is strong inherent relationship between the characteristics of chemical compounds and their molecular structures from a large number of experimental studies.The molecular structure of a chemical compound can be represented as a graph,which is called a molecular graph.A topological index is a mapping from molecular graphs to the set of real numbers.In order to describe the physicochemical properties,pharmaceutical properties,biological properties and other properties,theoretical chemists and mathematicians introduced and investigated various topological indices.The Mostar index Mo(G)of a graph G was introduced by Doslic et al.in 2018 as Mo(G)=(?)|nu-nv|,where nu is the number of vertices of G closer to vertex u than to vertex v,and nv is the number of vertices closer to vertex v than to vertex u.The Mostar index Mo(G)of a graph G is a global measure of the periphery of graph G and also measures how far is graph G from being distance-balanced graphs.The first and second Zagreb indices of a graph G are defined as where d(x)denotes the degree of a vertex x in G.The difference of Zagreb indices of the graph G is defined as ?M(G)=M2(G)-M1(G).From a mathematical point of view,two main problems on the study of Mostar index and the difference of Zagreb indices are as follows:(1)How to calculate the maximum and minimum values of these two parameters?(2)How to find the related extremal graphs?This PhD dissertation focuses on the study of Mostar index and the difference of Zagreb indices for some special graphs.We make use of transformation and structural analysis of graphs,optimization theory and other methods.Moreover,several topological indices of some chemical drugs are calculated.The PhD dissertation consists of four chapters as follows.In Chapter 1,we introduce the basic notations and terminology related to this dissertation and give a detailed survey in the related fields.And we give the main results obtained in the dissertationIn Chapter 2,we study the Mostar index of two kinds of graphs.First,the upper bounds and extremal graphs of Mostar index for cacti we derive.Next,we establish the upper and lower bounds of Mostar index for maximal planar graphs with minimum degree four and diameter two,and find the corresponding extremal graphsIn Chapter 3,we establish the tight upper and lower bounds of the difference of Zagreb indices for maximal planar graphs with minimum degree four and diameter two,extremal graphs attaining these upper and lower bounds are characterized.Furthermore,we study the difference of Zagreb indices for general Halin graphs and special Halin graphs with given number of inner verticesIn Chapter 4,as an application,we focus on Hyaluronic Acid-Paclitaxel conjugates by determining certain topological indices.