The Eccentric Distance Sum, the Harary Index, and the Degree Powers of Graphs with Given Diameter

Abstract

Let $G$ be a simple connected graph with the vertex set $V(G)$. The eccentric distance sum of $G$ is defined as ${\xi }^d(G)=\sum _{v\in V(G)}\epsilon (v)D_G(v)$, where $\epsilon (v)$ is the eccentricity of the vertex $v$ and $D_G(v)$ is the sum of all distances from the vertex $v$. The Harary index of $G$ is defined as $H(G)=\sum _{u,v\in V(G)}\frac{1}{d(u,v)}$, where $d(u,v)$ is the distance between $u$ and $v$ in $G$. The degree powers of $G$ is defined as $H(G)=\sum _{|u,v|\subseteq V(G)}\frac{1}{d(u,v)}$ for the natural number $p\ge 1$. In this paper, we determine the extremal graphs with the minimal eccentric distance sum, the maximal Harary index, and the maximal degree powers among all graphs with given diameter.