On Hyperbolic Sombor index of graphs

TL;DR

This paper corrects previous inaccuracies and extends the mathematical analysis of the Hyperbolic Sombor index, providing new bounds and characterizations for various graph classes, advancing understanding of this recently introduced topological index.

Contribution

It corrects errors in prior work on the Hyperbolic Sombor index and establishes new mathematical properties, bounds, and characterizations for different classes of graphs.

Findings

Corrected inaccuracies in earlier studies.

Derived bounds for the Hyperbolic Sombor index based on graph parameters.

Characterized extremal graphs attaining these bounds.

Abstract

The Hyperbolic Sombor index $HSO(G)$ of a graph $G$ is defined as \begin{align*} HSO(G) = \sum_{v_iv_j \in E(G)} \frac{\sqrt{d_i^{2}+d_j^{2}}}{\min\{d_i,d_j\}}, \end{align*} where $d_i$ and $d_j$ denote the degrees of the vertices $v_i$ and $v_j$, respectively. This index was recently introduced by Barman et al. [Geometric approach to degree-based topological index: Hyperbolic Sombor index, MATCH Commun. Math. Comput. Chem. 95 (2026) 63-94], who explored some of its mathematical properties and applications. However, their work contains several inaccuracies that require correction. In this paper, we first identify and rectify the errors found in the earlier study. We then extend the investigation by establishing new mathematical results for the Hyperbolic Sombor index across various classes of graphs, including trees, unicyclic graphs, and bicyclic graphs. In addition, we derive some lower and upper bounds for $HSO(G)$ in terms of the number of edges, maximum degree and minimum degree, and we characterize the graphs that attain these bounds. Finally, we conclude the paper by outlining potential directions for future research in this emerging area.