On the hyperbolic Sombor index and its counterpart

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} 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.