Multiplicatively weighted Harary index of some graph operations

Abstract

Recently, ALIZADEH et al proposed a modification of the Harary index in which the contributions of vertex pairs were weighted by the product of their degrees. It is named multiplicatively weighted Harary index and defined as: \begin{document}${H_M}\left( G \right) = \sum\limits_{u \ne v} {\frac{{{\delta _G}\left( u \right){\delta _G}\left( v \right)}}{{{d_G}\left( {u,v} \right)}}} $\end{document} , where δG(u) denotes the degree of the vertex u in the graph G and dG(u, v) denotes the distance between two vertices u and v in the graph G. In this paper, the explicit formulae for the multiplicatively weighted Harary index of tensor product G×Kr, the strong product G Kr and the wreath product G1oG2 in terms of other graph invariants including additively weighted Harary index, Harary index, the first and the second Zagreb indices and the first and the second Zagreb coindices, are obtained, where Kr is the complete graph. Additionally, we apply our results to compute the multiplicatively weighted Harary index of open fence and closed fence graphs.