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Theta-Relations Among Degree-Based Tree Indices

Duaa Abdullah

Abstract

Degree-based topological indices enable structural analysis of graphs, with key applications in chemical graph theory. This paper explores connections among three tree indices: Albertson, Sombor, and Sigma. The Sigma index tightly bounds the Sombor index via sharp two-sided inequalities, revealing their asymptotic equivalence up to constants. A pure $Θ$-relationship links Sombor and Albertson indices for extremal trees with fixed degree sequences, showing quadratic degree interactions scale with absolute degree disparities. These results position the Sombor index as an intermediate descriptor, bridging global degree dispersion and local edge irregularity in trees.

Theta-Relations Among Degree-Based Tree Indices

Abstract

Degree-based topological indices enable structural analysis of graphs, with key applications in chemical graph theory. This paper explores connections among three tree indices: Albertson, Sombor, and Sigma. The Sigma index tightly bounds the Sombor index via sharp two-sided inequalities, revealing their asymptotic equivalence up to constants. A pure -relationship links Sombor and Albertson indices for extremal trees with fixed degree sequences, showing quadratic degree interactions scale with absolute degree disparities. These results position the Sombor index as an intermediate descriptor, bridging global degree dispersion and local edge irregularity in trees.
Paper Structure (5 sections, 13 theorems, 72 equations, 3 figures)

This paper contains 5 sections, 13 theorems, 72 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Main Result
  5. Conclusion

Key Result

Proposition 2.1

Let $G$ be a graph with minimum degree $\delta$, maximum degree $\Delta$, and $n$ vertices. Then

Figures (3)

  • Figure 1: An example of tree $\mathcal{T}\in \mathcal{T}_{n,\Delta}$.
  • Figure 2: Lower and upper bound in Proposition \ref{['fibpron1task']}.
  • Figure 3: The impact of $\mathrm{SO}(\mathcal{T})$, $\mathrm{irr}(\mathcal{T})$ and $M(\mathcal{T})$.

Theorems & Definitions (22)

  • Definition 1: Zagreb Indices gutman1972totalgutman1975acyclic
  • Proposition 2.1: dorjsembe2022irregularity
  • Lemma 2.1: fath2013extremelygutman2018inverse
  • Theorem 2.2: Gutman2021Geo
  • Theorem 2.3: Gutman2021GeoLiu2023GutmanYou
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.1
  • ...and 12 more