Multiplicatively weighted Harary index of some composite graphs

Recently, Alizadeh et al. [Discrete Math., 313 (2013): 26-34] proposed a modification of the Harary index in which the contributions of vertex pairs are weighted by the product of their degrees. It is named multiplicatively weighted Harary index and defined as: $H_{M}(G)=\sum_{u\neq v}\frac{\delta_{G}(u)\cdot \delta_{G}(v)}{d_{G}(u,v)},$ where $\delta_{G}(u)$ denotes the degree of the vertex $u$ in the graph $G$ and $d_{G}(u,v)$ denotes the distance between two vertices $u$ and $v$ in the graph $G.$ In this paper, after establishing basic mathematical properties of this new index, we proceed by finding the extremal graphs and presenting explicit formulae for computing the multiplicatively weighted Harary index of the most important graph operations such as the join, composition, disjunction and symmetric difference of graphs.