Multiplicatively weighted Harary index of graphs
Let \(G\) be a connected graph with vertex set \(V(G)\). The multiplicatively weighted Harary index of a graph \(G\) is defined as \(H_M(G)=\sum _{u\ne v}\frac{d(u)d(v)}{d(u,v)}\), where \(d(u)\) is the degree of vertex \(u\), and \(d(u,v)\) denotes the distance between \(u\) and \(v\). In this paper, we first prove that the multiplicatively weighted Harary index of a graph is monotonic on some transformations, and then determine the extremal values of the multiplicatively weighted Harary indices for some familiar classes of graphs and characterize the corresponding extremal graphs.
multiplicatively weighted harary index harary index degree distance tree unicyclic graph 05c07 05c15 05c50