The Hyper-Wiener Index of Unicyclic Graphs with Given Matching Number

Abstract

Let \(G\) be a connected simple graph. The hyper-Wiener index \(WW(G)\) is defined as \(WW(G) = \sum_{u,v \in V(G)} (d(u, v) + d^2(u,v)),\) with the summation going over all pairs of vertices in \(G\). In this paper, we determine the extremal unicyclic graphs with given matching number and minimal hyper-Wiener index.