Further results on the reciprocal degree distance of graphs

The reciprocal degree distance of a simple connected graph (G=(V_G, E_G)) is defined as (bar{R}(G)=sum _{u,v in V_G}(delta _G(u)+delta _G(v))frac{1}{d_G(u,v)}), where (delta _G(u)) is the vertex degree of (u), and (d_G(u,v)) is the distance between (u) and (v) in (G). The reciprocal degree distance is an additive weight version of the Harary index, which is defined as (H(G)=sum _{u,v in V_G}frac{1}{d_G(u,v)}). In this paper, the extremal (bar{R})-values on several types of important graphs are considered. The graph with the maximum (bar{R})-value among all the simple connected graphs of diameter (d) is determined. Among the connected bipartite graphs of order (n), the graph with a given matching number (resp. vertex connectivity) having the maximum (bar{R})-value is characterized. Finally, sharp upper bounds on (bar{R})-value among all simple connected outerplanar (resp. planar) graphs are determined.