On the Relation Between Wiener Index and Eccentricity of a Graph

TL;DR

This paper explores the relationship between the Wiener index and eccentricity of graphs, establishing bounds, extremal cases, and conjectures, with specific focus on trees and their structural properties.

Contribution

It provides new bounds, characterizations, and formulas relating Wiener index and eccentricity, including extremal graphs and conjectures about their behavior under graph operations.

Findings

Established bounds on Wiener index in terms of eccentricity.

Characterized extremal graphs for these bounds.

Proposed two conjectures about the behavior of the Wiener index and eccentricity.

Abstract

The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type result on $W(G)$ involving $\varepsilon(G)$ is given. A sharp upper bound on the Wiener index of a tree in terms of its eccentricity is proved. It is shown that in the class of trees of the same order, the difference $W(T) - \varepsilon(T)$ is minimized on caterpillars. An exact formula for $W(T) - \varepsilon(T)$ in terms of the radius of a tree $T$ is obtained. A lower bound on the eccentricity of a tree in terms of its radius is also given. Two conjectures are proposed. The first asserts that the difference $W(G) - \varepsilon(G)$ does not increase after contracting an edge of $G$. The second conjecture asserts that the difference between the Wiener index of a graph and its eccentricity is largest on paths.