Theta-Relations Among Degree-Based Tree Indices

TL;DR

This paper explores the relationships among degree-based topological indices in trees, establishing bounds and equivalences that clarify how these indices reflect structural irregularities and degree deviations.

Contribution

It introduces new bounds and relationships among the Albertson, Sombor, and Sigma indices, revealing their asymptotic and extremal connections in trees.

Findings

The Sigma index tightly controls the Sombor index in trees.

Sombor and Sigma indices are asymptotically equivalent up to constants.

A Theta-relationship between Sombor and Albertson indices is established.

Abstract

In this paper, degree-based topological indices play a key role in the structural analysis of graphs in this paper and have significant uses in chemical graph theory. We investigate the connections between three such tree indices: the Albertson, Sombor, and Sigma indices. We show that the quadratic degree deviation, measured by the Sigma index, tightly controls the Sombor index of a tree by establishing sharp two-sided bounds. We demonstrate that the Sombor and Sigma indices are asymptotically equivalent up to constant factors as a direct result. A pure $\Theta$-relationship between the Sombor index and the Albertson index is derived by taking into account extremal trees with a fixed degree sequence. This finding demonstrates that, in extremal configurations, quadratic degree interactions and absolute degree disparities scale appropriately. Overall, our data suggest that the Sombor index functions as an intermediate descriptor, capturing both global degree dispersion and local edge irregularity. From a structural standpoint, these findings clarify the relationship between vertex-based and edge-based irregularity measurements in trees.