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.