The comparison of two Zagreb-Fermat eccentricity indices

TL;DR

The paper investigates the relationship between the first and second Zagreb-Fermat eccentricity indices, $F_1(G)$ and $F_2(G)$. It proves that $F_2(G)/m(G) \le F_1(G)/n(G)$ for all acyclic and unicyclic graphs by leveraging structural decompositions: trees via diametrical paths and central-vertex structure, and unicyclic graphs via a cycle-plus-attached-trees framework. The authors also provide explicit counterexamples showing that the inequality does not hold in general for multicyclic graphs, highlighting distinct extremal behaviors across graph classes. This work advances the understanding of Zagreb-Fermat indices and their extremal properties, with implications for graph-structural analysis in chemical graph theory and network design.

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.

Figures (7)

• Figure 1: $G$ expanding at $P_{d+1}$ (One central vertex case).
• Figure 2: $G$ expanding at $P_{d+1}$ (Two central vertices case).
• Figure 3: Unicyclic graph $G$.
• Figure 4: $T_x$ expanding at $P_{d+1}$.
• Figure 5: $T_x$ expanding at $P_{d+1}$. Here let $x=v_{s+1}$ for notational convenience.

Theorems & Definitions (19)

• Lemma 1
• proof
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof