The comparison of two Zagreb-Fermat eccentricity indices

Xiangrui Pan, Longyu Li, Gengji Li, Cheng Zeng

Published 2023-09-22Version 1

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.