Multiplicatively weighted Harary index of some composite graphs

Abstract

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.