On the extremal properties of the average eccentricity

TL;DR

This paper investigates the extremal values of the average eccentricity in graphs, introduces transformations affecting it, and resolves multiple conjectures related to graph parameters, advancing understanding of graph structure properties.

Contribution

It introduces two graph transformations that alter average eccentricity and resolves four conjectures related to eccentricity and other graph invariants.

Findings

Identified transformations that increase or decrease average eccentricity.

Resolved four conjectures about eccentricity, clique number, Randić index, and independence number.

Refuted one conjecture regarding average eccentricity and minimum vertex degree.

Abstract

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc (G)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease $ecc (G)$. Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number, the Randi\' c index and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.