Research On Some Topological Index Of Graphs

Topological index is an invariant of graphs,which can be used to de-scribe chemical structures.It is also widely used in quantitative structure-property/activity correlation models.Topological index based on the distance between vertices of graphs is widely used to predict physical properties,chem-ical properties and biological activities of compounds in chemical theory.The Wiener index of graphs is the most classical topological index,which is based on the distance between vertices of graphs.It was proposed by Wiener in 1947.In 1993,Plavsic and Ivanciuc introduced the Harary index of graphs.Randic proposed the hyper-Wiener index of acyclic graphs in 1993.After that,Klein extended the definition of hyper-Wiener index to all connected graphs.The concept of resistance distance of graphs was proposed by Klein and Randic in 1993.Recently,we defined a new topological index based on resistance distance,the reciprocal degree resistance distance index.In this paper,we give some sufficient conditions for a coonnected graph with given minimum degree to be Hamilton-connected and from every vertex to traceable,in terms of the Wiener index,hyper-Wiener index,Harary in-dex of the complement of graphs,respectively.And we determine the graph with maximum reciprocal degree resistance distance index among all uni-cyclic graphs and bicyclic graphs,and characterize the corresponding extremal graph.The paper is arranged as follows:In the chapter 1,we first give the background of this study,then give the basic symbols and concepts,the last give the research status and main conclu-sions of other properties such as Hamilton of Wiener-type index of graphsIn the Chapter 2,we first give the sufficient condition of the number of edges for a graph to be Hamilton-connected and from every vertex to traceable,according to the conversion of quantitative relations between the number of edges and the topological index of graphs,give give some sufficient conditions for a coonnected gra.ph with given minimum degree to be Hamilton-connected and from every vertex to traceable,in terms of the Wiener index,hyper-Wiener index,Harary index of the complement of graphsIn the Chapter 3,we first give the formula for calculating the resistance distances of any two vertices on a circle,then give the lemmas of edge-lifting transformation of graphs,and cycle-lifting transformation and cycle-shrinking transformation of unicyclic graphs and bicyclic graphs,after that we determine the graph with maximum reciprocal degree resistance distance index among all unicyclic graphs and bicyclic graphs and characterize the corresponding extremal graph.