The comparison of two Zagreb-Fermat eccentricity indices

In this paper, we focus on comparing the first and second Zagreb-Fermat eccentricity indices of graphs. We show that $$\frac{\sum _{uv\in E\left(G\right)}{\epsilon }_3\left(u\right){\epsilon }_3\left(v\right)}{m\left(G\right)}\le \frac{\sum _{u\in V\left(G\right)}{\epsilon }_3^2\left(u\right)}{n\left(G\right)}$$ holds for all acyclic and unicyclic graphs. Besides, we verify that the inequality may not be applied to graphs with at least two cycles.