The comparison of two Zagreb-Fermat eccentricity indices

TL;DR

This paper compares two Zagreb-Fermat eccentricity indices in graphs, establishing inequalities for acyclic and unicyclic graphs and highlighting limitations for graphs with multiple cycles.

Contribution

It introduces a comparison inequality between the two indices for specific graph classes and identifies cases where the inequality does not hold.

Findings

Inequality holds for acyclic and unicyclic graphs.

Inequality does not generally apply to graphs with multiple cycles.

Provides mathematical proof for the inequality in specific graph classes.

Abstract

In this paper, we focus on comparing the first and second Zagreb-Fermat eccentricity indices of graphs. We show that $$\frac{\sum_{uv\in E\left( G \right)}{\varepsilon_3\left( u \right) \varepsilon_3\left( v \right)}}{m\left( G \right)} \leq \frac{\sum_{u\in V\left( G \right)}{\varepsilon_{3}^{2}\left( u \right)}}{n\left( G \right)} $$ holds for all acyclic and unicyclic graphs. Besides, we verify that the inequality may not be applied to graphs with at least two cycles.