  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.
