On the Hyperbolic Sombor Index and Its Counterpart

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.