The Topological Index And Some Properties Of Graphs

The topological index is a method to numerate molecular structure.It is achieved by performing a numeric operation on the matrix of representing the molec-ular graph.The Wiener index of graphs is one of the topological indices that have been studied in depth.It was put forward by Wiener in 1947,indicating that the sum of distances between all molecules.It is not only an important concept in pure graphics,but also related to the physical and chemical properties of various chem-ical compounds.In 1993,Plavsic et al.introduced the Haraxy index of graphs,which referred to the sum of reciprocal of distances between all molecules.Randie proposed the hyper-Wiener index of acyclic graph at the same year.Then Klein ex-tended the definition of hyper-Wiener index to all connected graphs.Wiener index,hyper-Wiener index and Harary index of graphs belong to Wiener type invariants of graphs.They are significant topological indexes studied in this paper.The com-mon molecular topological indexes include also Balaban index,Randic-Kier index,Hosoya index,Kovats index,Zagrb index,Schultz,etc.The well-known Hamilton problem is to judge whether a given undirected graph contains a Hamilton cycle.However,no ideal method has been found up to now.Thus,many scholars pursue new ways to solve this problem.Because the topologi-cal index of graphs can well reflect the structural properties of graphs and facilitate computation.Recently,people have begun to study the Hamilton problem of graphs through topological index.It opens up a new way to solve the Hamilton problem(NP-complete problem).This paper studies the Hamilton problem and mainly de-picts the Hamiltonian of general graph,balance bipartite graph,nearly balanced bipartite graph and k-connected graph by utilizing the Wiener index,hyper-Wiener index and Haxary index of graphs and their complements.The paper also through utilizing hyper-Wiener index of graphs discusses that the graph is k-path-coverable,k-Hamiltonian,k-edge-Hamiltonian,β-debate,etc.The main contents are arranged as follows:In Chapter 1,we introduce the background and significance of the research,the research status of topological index,some properties of graphs,related symbols and basic concepts,and then offer the whole structure of the paper;In Chapter 2,firstly,we give sufficient conditions of the general graph to be traceable and Hamiltonian by utilizsing the Wiener index,hyper-Wiener index and Harary index of the complement.Secondly,with the hyper-Wiener index of graphs,we give the sufficient conditions for the graph to be k-connected,β-debate,k-hamilton,k-path-covered and k-side-hamilton;In Chapter 3,we give some sufficient conditions for a balanced bipartite graph to be traceable by utilizing the Wiener index,hyper-Wiener index and Harary index of their quasi-complement;In Chapter 4,we give some sufficient conditions for a nearly balanced bipartite graph to be traceable by using the Wiener index,hyper-Wiener index and Harary index of graph and its quasi-complement;In Chapter 5,we give some sufficient conditions for a k-connected graph to be Hamilton-connected and traceable for every vertex by using the Wiener index,hyper-Wiener index and Harary index of graph and its quasi-complement.