A Sharp Lower Bound for the Wiener Index of a Graph

Abstract

Given a simple connected undirected graph \(G\), the Wiener index \(W(G)\) of \(G\) is defined as half the sum of the distances over all pairs of vertices of \(G\). In practice, \(G\) corresponds to what is known as the molecular graph of an organic compound. We obtain a sharp lower bound for \(W(G)\) of an arbitrary graph in terms of the order, size, and diameter of \(G\).