On Extremal Family Trees $(\mathcal{T}_n)_{n\geqslant 3}$ Beyond Caterpillars and Greedy Constructions

TL;DR

The paper analyzes the sigma index, a quadratic irregularity measure $\sigma$, on the class of trees and their degree-sequence realizations. It proves that among all trees realizing a degree sequence, the greedy tree minimizes $\sigma$, whereas caterpillars do not achieve the global minimum; it then derives a closed-form expression for $\sigma$ on a spine-and-pendant tree, revealing additive level-wise contributions. It demonstrates that non-caterpillar, non-greedy trees can have $\sigma$-values strictly between the global minimum and the caterpillar minimum, highlighting limitations of the two standard subfamilies. The results provide structural insight and a framework for comparing linear and quadratic irregularity across tree families, with potential implications in chemical graph theory and network topology.

Abstract

This paper investigates topological indices for the greedy tree $\mathcal{T}_\mathscr{D}$ associated with a graphic degree sequence $\mathscr{D} = (d_1 \geqslant d_2 \geqslant \dots \geqslant d_n)$ of a tree. A fundamental challenge in the study of topological indices lies in establishing precise bounds, as such findings illuminate intrinsic relationships among diverse indices. We investigate the extremal properties of the graph invariant $σ$ over the family $\mathcal{T}_n$ of all trees on $n \ge 3$ vertices. Specifically, we compare the minimum values of $σ$ attained in restricted subclasses -- including caterpillar trees and greedy trees -- with the global minimum over $\mathcal{T}_n$. We prove that caterpillar trees do not achieve the minimum value of $σ$ among all trees, whereas greedy trees attain values no smaller than this global minimum. Moreover, we show that certain trees, which are neither caterpillars nor greedy trees, have $σ$-values strictly between the global minimum over $\mathcal{T}_n$ and the minimum among caterpillar trees. These results highlight structural limitations of these common tree classes in extremal problems and offer new insights into the role of non-caterpillar, non-greedy trees in minimizing graph invariants.

Figures (4)

• Figure 1: Caterpillar tree.
• Figure 2: A greedy tree.
• Figure 3: Caterpillar tree $\mathbb{C}(n,p)$.
• Figure 4: The behavior of caterpillar tree $\mathbb{C}(n,p)$.

Theorems & Definitions (28)

• Definition 1: Caterpillar tree
• Definition 2: Greedy tree
• Lemma 2.1: gutman2018inverse
• Lemma 2.2: yang2023sigma
• Theorem 2.3: HamoudwithDuaa
• Theorem 2.4: HamoudwithDuaa
• Theorem 2.5: gutman2018inverse
• Lemma 2.6: Sun2025DuJMei
• Proposition 3.1
• proof