Further results on the reciprocal degree distance of graphs

Abstract

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.