Proof of an open problem on the Sombor index

TL;DR

This paper determines extremal graphs for the Sombor index under connectivity constraints, resolves an open problem, provides counterexamples to previous claims, and explores the index's application in QSPR modeling.

Contribution

It solves an open problem by identifying extremal graphs for the Sombor index under connectivity constraints and offers new insights with counterexamples and QSPR analysis.

Findings

Maximal and minimal graphs identified for given connectivity constraints.

Counterexamples provided for previous claims in the literature.

Sombor index effectively used in QSPR regression models.

Abstract

The Sombor index is one of the geometry-based descriptors, which was defined as $$SO(G)=\sum_{uv\in E(G)}\sqrt{d^{2}(u)+d^{2}(v)},$$ where $d(u)$ (resp. $d(v)$) denotes the degree of vertex $u$ (resp. $v$) in $G$. In this note, we determine the maximum and minimum graphs with respect to the Sombor index among the set of graphs with vertex connectivity (resp. edge connectivity) at most $k$, which solves an open problem on the Sombor index proposed by Hayat and Rehman [On Sombor index of graphs with a given number of cut-vertices, MATCH Commun. Math. Comput. Chem. 89 (2023) 437--450]. For some of the conclusions of the above paper, we give some counterexamples. At last, we give the QSPR analysis with regression modeling and Sombor index.