On the multiplicatively weighted Harary index of composite graphs

Let HM(G) be the multiplicatively weighted Harary index of the molecular graph G, which is defined as HM(G)=∑u,vV(G)dG(u)dG(v)dG(u,v), where dG(u) is the degree of a vertex u∈V(G), and the dG(u,v) denotes the distance between u and v in G. We introduce four graph operations and obtain explicit formulas for the values of multiplicatively weighted Harary index of composite graphs generated by the four graph operations. Based on this, a lower and an upper bound is determined for the multiplicatively weighted Harary index among graphs in each of the four classes of composite graphs.