On The Multiplicatively Weighted Harary Index Of Some Composite Graphs

It is known that the intuitive idea of pairs of close atoms contributing more than the distant ones has been difficult to capture in topological indices.So the multiplica-tive weighted Harary index was prorosed by Alizadeh.It is defined that HM(G)=(?)dG(u,v)/dG(u)dG(v),where dG(u)is the degree of a vertex u ? V(G),and the dis-tance dG(u,v)between vertices u and v in G is the length of any shortest path in G connecting u and v.In this paper we mainly introduce the four graph operations resulting in composite graphs,present explicit formulas for the values of multiplicative weighted Harary index of them,and determine a lower and an upper bound for the multiplicative weighted Harary index of these composite graphs.The main contents as follows:In the first chapter,we introduce the research background,the research progress,as well as the main research contents of this article.Some preliminary knowledge are also given in this chapter.In the second chapter,by introducing the four standard graph operations resulting in composite graphs,we present explicit formulas for the values of multiplicative weighted Harary index of them.In the third chapter,we determine a lower and an upper bound for these composite graphs.In the fourth chapter,we summarize the main results of this paper,and put forward the problems of further research.