Further results on the reciprocal degree distance of graphs

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