On the Hyperbolic Sombor Index and Its Counterpart

TL;DR

This paper studies the hyperbolic Sombor index and its counterpart, establishing conditions for how these indices change with edge addition, providing bounds, and correcting previous inaccuracies in the literature.

Contribution

It corrects inaccuracies from prior work, establishes conditions for index changes upon edge addition, and derives bounds for the hyperbolic Sombor and CDSO indices.

Findings

Conditions for $HSO(G+vw) > HSO(G)$ and $HSO(G+vw) < HSO(G)$ are established.

A lower bound on $HSO(G)$ based on graph order and size is provided.

Results are extended to the complementary diminished Sombor (CDSO) index.

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.