The average eccentricity of a graph with prescribed girth

TL;DR

This paper establishes upper bounds on the average eccentricity of connected graphs based on their order, minimum degree, and girth, and explores the sharpness of these bounds through graph constructions.

Contribution

It provides new upper bounds on average eccentricity considering girth and degree, and demonstrates their asymptotic sharpness via graph constructions.

Findings

Upper bounds on average eccentricity in terms of order, degree, and girth.

Construction of graphs showing bounds are asymptotically sharp.

Improved bounds for graphs with large maximum degree.

Abstract

Let $G$ be a connected graph of order $n$. The eccentricity $e(v)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the mean of all eccentricities in $G$. We give upper bounds on the average eccentricity of $G$ in terms of order $n$, minimum degree $\delta$, and girth $g$. In addition, we construct graphs to show that, if for given $g$ and $\delta$, there exists a Moore graph of minimum degree $\delta$ and girth $g$, then the bounds are asymptotically sharp. Moreover, we show that the bounds can be improved for a graph of large degree $\Delta$.