On edge-weighted mean eccentricity of graphs

TL;DR

This paper investigates bounds on the edge-weighted mean eccentricity of connected graphs, providing new theoretical limits based on graph parameters and exploring Nordhaus-Gaddum-type results.

Contribution

It introduces bounds on weighted mean eccentricity for trees and non-tree graphs, and extends results to Nordhaus-Gaddum-type inequalities.

Findings

Derived upper and lower bounds for weighted mean eccentricity.

Established bounds in terms of graph order, size, and edge-connectivity.

Presented Nordhaus-Gaddum-type results for the weighted eccentricity.

Abstract

Let $G$ be a connected edge-weighted graph of order $n$ and size $m$. Let $w:E(G)\rightarrow \mathbb{R}^{\geq 0}$ be the weighting function. We assume that $w$ is normalised, that is, $\sum_{e\in E(G)} w(e)=m$. The weighted distance $d_w(u,v)$ between any two vertices $u$ and $v$ is the least weight between them and the eccentricity $e_w(v)$ of a vertex $v$ is the weighted distance from $v$ to a vertex farthest from it in $G$. The mean(average) eccentricity of $G$, $avec(G,w)$, is the (weighted) mean of all eccentricities in $G$. We obtain upper and lower bounds on $avec(G,w)$ in terms of $n$, $m$ or edge-connectivity $\lambda$ for two cases: $G$ is a tree and $G$ is connected but not a tree. In addition, we obtain the Nordhaus-Gaddum-type results for edge-weighted average eccentricity.