On the Hyperbolic Sombor Index and Its Counterpart
Abeer M. Albalahi, Shibsankar Das, Akbar Ali, Jayjit Barman, Amjad E. Hamza
TL;DR
The paper investigates the hyperbolic Sombor index $HSO(G)$ and its complementary diminished counterpart $^cDSO(G)$, aiming to correct inaccuracies in prior work and to understand how edge insertions affect these degree-based topological descriptors. It derives tight extremal bounds and conditions, including $HSO(G) \ge \sqrt{2}\,m$ with equality for regular $G$ and relations to the classical Sombor index via $(1/\Delta) SO(G) \le HSO(G) \le (1/\delta) SO(G)$, plus cycle/star extremals $HSO(C_n) \le HSO(G) \le HSO(S_n)$ for $n\ge3$. The authors also establish analogous results for $^cDSO(G)$, such as $^cDSO(G) \le m\sqrt{2}$ with equality for regular graphs and global bounds $\sqrt{(n-1)^2+1} \le ^cDSO(G) \le n(n-1)/\sqrt{2}$, and they analyze how adding edges $G+vw$ can increase or decrease these indices under certain degree conditions. A key contribution is the careful correction and strengthening of previous theorems, together with a unified set of extremal results and edge-addition analyses that deepen understanding of how degree distributions govern these indices in chemical graphs.
Abstract
For a graph $G$ with edge set $E$, let $d(w)$ denote the degree of a vertex $w$ in $G$. The hyperbolic Sombor index of $G$ is defined by $$HSO(G)=\sum_{uv\in E}(\min\{d(u),d(v)\})^{-1}\sqrt{(d(u))^2+(d(v))^2}.$$ If $\min\{d(u),d(v)\}$ is replaced with $\max\{d(u),d(v)\}$ in the formula of $HSO(G)$, then the complementary diminished Sombor (CDSO) index is obtained. For two non-adjacent vertices $v$ and $w$ of $G$, the graph obtained from $G$ by adding the edge $vw$ is denoted by $G+vw$. In this paper, we attempt to correct some inaccuracies in the recent work [J. Barman, S. Das, Geometric approach to degree-based topological index: hyperbolic Sombor index, MATCH Commun. Math. Comput. Chem. 95 (2026) 63-94]. We establish a sufficient condition under which $HSO(G+vw) > HSO(G)$ holds, and also provide a sufficient condition guaranteeing $HSO(G+vw) < HSO(G)$. In addition, we give a lower bound on $HSO(G)$ in terms of the order and size of $G$. Furthermore, we obtain similar results for the CDSO index.