On the Wiener Index of Unicyclic Graphs with Given Girth

Abstract

The Wiener index of a graph \(G\) is defined as \(W(G) = \sum_{u,v \in V(G)} d_G(u,v),\) where \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\) and the sum goes over all pairs of vertices. In this paper, we investigate the Wiener index of unicyclic graphs with given girth and characterize the extremal graphs with the minimal and maximal Wiener index.