Reciprocal degree distance of some graph operations

The reciprocal degree distance (RDD)‎, ‎defined for a connected graph G as vertex-degree-weighted sum of the reciprocal distances‎, ‎that is‎, ‎RDD(G) =\sum\limits_{u,v\in V(G)}\frac{d_G(u)‎ + ‎d_G(v)}{d_G(u,v)}. The reciprocal degree distance is a weight version of the Harary index‎, ‎just as the degree distance is a weight version of the Wiener index‎. ‎In this paper‎, ‎we present exact formulae for the reciprocal degree distance of join‎, ‎tensor product‎, ‎strong product and wreath product of graphs in terms of other graph invariants including the degree distance‎, ‎Harary index‎, ‎the first Zagreb index and first Zagreb coindex‎. ‎Finally‎, ‎we apply some of our results to compute the reciprocal degree distance of fan graph‎, ‎wheel graph‎, ‎open fence and closed fence graphs‎.

Keywords:

Reciprocal degree distance , Harary index , Graph operations