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

TL;DR

This paper proves a conjecture stating that contracting any edge in a connected graph with at least three vertices reduces the difference between the Wiener index and eccentricity, extending previous results beyond bridges.

Contribution

It confirms the conjecture for all connected graphs with at least three vertices, regardless of the edge type, generalizing prior partial proofs.

Findings

The difference between Wiener index and eccentricity decreases after edge contraction.

The conjecture holds for all connected graphs with at least three vertices.

The result extends previous proofs limited to bridge edges.

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 $\varepsilon(G)$, 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 $\varepsilon(G)$ 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.