On Topological Indices Based On Eccentricity Of Graphs

In the fields of chemical graph theory,molecular topology and mathematical chemistry,topological index(also known as connectivity index)is a kind of molecular descriptor,which is calculated from the molecular graph of a compound.Topological index is a digital parameter of graph transformation.Topological index is usually used in the development of quantitative structureactivity relationship(QSAR),in which the biological activity or other properties of a molecule is related to its chemical structure,it has received attention from many scholars and has developed rapidly.Wiener index is named after Harry Wiener,which was introduced in 1947.It is the oldest topological index related to molecular branching,and Wiener index is a topological index based on distance.The eccentricity of a vertex in the graph is defined as the maximum distance from this vertex to other vertices in the graph.In 1997,Sharma proposed the eccentric connectivity index,which caused many scholars to study the problems related to eccentricity.This paper mainly studies the related problems of the topological index based on the eccentricity of the graph.The full text is divided into five chapters,as follows:In Chapter 1,we introduce the basic concepts and symbols of graph theory,the definition of index and the current research status.In Chapter 2,we study the extreme values of the total eccentricity index of the maximal outerplanar graphs.Using mathematical induction and graph transformation,the maximum values of the maximal outerplanar graphs with9)vertices with respect to the total eccentricity index are determined,and the corresponding extreme value graphs are obtained.In Chapter 3,we study the properties of the leap eccentric connectivity index of graphs and the graph operation based on subdivided edges.Firstly,the upper and lower bounds of the leap eccentric connectivity index are expressed by using other topological indices related to the eccentricity,and some results about the complement graph are given.Then,the upper and lower bounds of the leap eccentric connectivity index of four graphs based on subdivision edges are characterized.Then,the expressions and bounds of the the leap eccentric connectivity index of the join graph based on subdivision and the four variants of the corona graph are given.In Chapter 4,we study the graph operations of eccentric harmonic index.The lower bounds of eccentric harmonic index of the five graph operations of cartesian product,composition,symmetric difference,join graph and corona graph of two graphs are described.In Chapter 5,we summarize the main work of this thesis and raise some prospects for future work.