The Hyper-Wiener Index of Trees with Given Parameters

Abstract

Let $G$ be a connected graph. The hyper-Wiener index $WW(G)$ is defined as $WW(G)=\frac{1}{2}\sum _{u,v\in V(G)}d(u,v)+\frac{1}{2}\sum _{u,v\in V(G)}{d}^2(u,v),$ with the summation going over all pairs of vertices in $G$ and $d(u,v)$ denotes the distance between $u$ and $v$ in $G$. In this paper, we determine the upper or lower bounds on hyper-Wiener index of trees with given number of pendent vertices, matching number, independence number, domination number, diameter, radius, and maximum degree.