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On Hyperbolic Sombor index of graphs

Kinkar Chandra Das, Sultan Ahmad

TL;DR

The paper addresses inaccuracies in the initial development of the Hyperbolic Sombor index $HSO(G)$ and provides corrected equality conditions and proofs, extending the analysis to trees, unicyclic, and bicyclic graphs. It establishes tight extremal bounds for unicyclic and bicyclic graphs, identifies exact extremal graphs such as $C_n$, $S'_n$, $C'_n$, $C''_n$, $S''_n$, and develops general edge-based bounds linking $HSO(G)$ to the Sombor index $SO(G)$ via $1/\Delta$ and $1/\delta$. Additionally, it derives universal bounds in terms of $m$, $\Delta$, and $\delta$, and proposes open problems and a conjecture regarding the maximality of $HSO$ for certain graph classes. These results deepen the theoretical understanding of $HSO$ and provide precise tools for evaluating degree-based hyperbolic descriptors in chemical and network graphs. $HSO(G)$ remains a promising descriptor for structural analysis, with corrected foundations enabling robust applications and further research.

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.

On Hyperbolic Sombor index of graphs

TL;DR

The paper addresses inaccuracies in the initial development of the Hyperbolic Sombor index and provides corrected equality conditions and proofs, extending the analysis to trees, unicyclic, and bicyclic graphs. It establishes tight extremal bounds for unicyclic and bicyclic graphs, identifies exact extremal graphs such as , , , , , and develops general edge-based bounds linking to the Sombor index via and . Additionally, it derives universal bounds in terms of , , and , and proposes open problems and a conjecture regarding the maximality of for certain graph classes. These results deepen the theoretical understanding of and provide precise tools for evaluating degree-based hyperbolic descriptors in chemical and network graphs. remains a promising descriptor for structural analysis, with corrected foundations enabling robust applications and further research.

Abstract

The Hyperbolic Sombor index of a graph 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 and denote the degrees of the vertices and , 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 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.

Paper Structure

This paper contains 5 sections, 10 theorems, 93 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Flaws in the results of BD1
  3. Extremal Results on Unicyclic and Bicyclic Graphs with Respect to the Hyperbolic Sombor Index
  4. Upper and Lower bounds on the Hyperbolic Sombor index of graphs
  5. Concluding Remarks

Key Result

Theorem 2.1

Let $G$ be a simple connected graph with maximum degree $\Delta$, minimum degree $\delta$, and Sombor index $SO(G)$. Then Moreover, the left equality holds if and only if $G$ is a regular graph, and the right equality holds if and only if $G$ is a regular graph or $G\in \Gamma$.

Figures (5)

  • Figure 1: Four graphs $H_1,\,H_2,\,H_3$, and $H_4$.
  • Figure 2: Two graphs $H_5$ and $H_6$.
  • Figure 3: Two graphs $S(a_1,a_2,a_3)$ and $S(n-3,0,0)\cong S'_n$.
  • Figure 4: Two graphs $C'_n$ and $C"_n$.
  • Figure 5: Two graphs $C_{3,3}$ and $S"_n$.

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Lemma 3.1
  • ...and 10 more