### 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.

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