2025/10/28 by Albalahi, Abeer M., Das, Shibsankar, Ali, Akbar +2
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2510.24809
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)=∑uv∈ E(min\d(u),d(v)\)-1√((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.