Proof of a Conjecture on Wiener Index and Eccentricity of a graph due to edge contraction

Abstract

For a connected graph $G$, the Wiener index, denoted by $W(G)$, is the sum of the distance of all pairs of distinct vertices and the eccentricity, denoted by $\epsilon (G)$, is the sum of the eccentricity of individual vertices. In \cite{Kc}, the authors posed a conjecture which states that given a graph $G$ with at least three vertices, the difference between $W(G)$ and $\epsilon (G)$ decreases when an edge is contracted and proved that the conjecture is true when $e$ is a bridge. In this manuscript, we confirm that the conjecture is true for any connected graph $G$ with at least three vertices irrespective of the nature of the edge chosen.