Regularities of Typical Sigma Index on Caterpillar Trees of Pendent Vertices

TL;DR

This paper investigates two key topological indices, the Albertson irregularity and the Sigma index, on caterpillar trees with varying spine and leaf configurations. It develops explicit closed-form expressions for these indices across several families of caterpillars, particularly focusing on the fundamental case C(n,3) and extending to general (n,m) constructions. Beyond Albertson and Sigma, the work also treats related indices such as Sombor, Randić, and sum-connectivity, providing both specific formulas and general frameworks to facilitate fast computation and extremal analysis. The results have implications for chemical graph theory and related computational tasks where precise index values enable rapid characterization and comparison of molecular structures modeled by caterpillar trees.

Abstract

In this paper, topological indices play a significant role in the analysis of caterpillar trees, especially due to their applications in chemical graph theory. We presented a study on topological indices related to the Sigma index, which we carefully selected on caterpillar trees with multiple levels. Through the election of these topological indices, we provide efficient models of these indices on caterpillar trees, as it is known that the Albertson index is the basis on which most of the topological indices are built and we have shown this through the close correlation between the Albertson's index and the Sigma index.

Figures (7)

• Figure 1: Example of Caterpillars $C(n,3)$
• Figure 2: The benzene molecule.
• Figure 3: General case of Caterpillar tree.
• Figure 4: Caterpillars tree with $(n,3)$ vertices
• Figure 5: Caterpillar tree of order $(3,4,5)$

Theorems & Definitions (43)

• Proposition 2.1
• Lemma 2.1: 1310
• Theorem 2.2: jas1
• Theorem 2.3: m1
• Definition 1: Chemical Molecules GrossYellen18
• Proposition 3.1
• proof
• Lemma 3.1
• Proposition 3.2
• Proposition 3.3