Research On Some Parameters Of Strong Product Graphs

Strong product is a kind of graph operation,which can be used to construct large graphs by using some small graphs.Large graphs retain many good properties of small graphs.In this paper,we mainly study some algebraic properties,topological structure parameters,Euler property and Hamiltonian property of strong product graphs.This paper adopts the following structure:In Chapter 1,we introduce the research background,current research status and the main work of this paper.In Chapter 2,we study the algebraic properties of strong product graphs,and prove the results that strong product graphs satisfy the commutative law,the associative law and the distributive law.In Chapter 3,we study the topological structures of strong product graphs.The vertex degree,edge number,distance,diameter,radius and forwarding index of strong product graphs are given.In Chapter 4,we study the Euler property of strong product graphs.We consider the existence of Euler tour and Euler entry in strong product graphs,and give some results.In Chapter 5,we study the Hamiltonian problem in strong product graphs.We first introduce a generalized form of Hamilton path and Hamilton cycle,that is,6)-distant Hamiltonian walks.Then we study the existence of 6)-distant Hamiltonian walks in strong product graphs.Finally,some sufficient conditions are given for strong product graphs to contain 6)-distant Hamiltonian walks.In Chapter 6,we give a summary of this paper and point out some directions for further research.