On Two Conjectures Concerning the Hyperbolic Sombor Index and Its Counterpart

The hyperbolic Sombor index (HSO index) and the complementary diminished Sombor index (CDSO index) of a graphwhere dG(u) and dG(v) are the degrees of the vertices u and v, respectively.Recently, two conjectures concerning the extremum values of the HSO and CDSO indices of connected graphs of fixed order and cyclomatic number were posed in [A.M. Albalahi, S. Das, A. Ali, J. Barman, A. E. Hamza, Discrete Math.Lett.16 (2025) 108-115].In this paper, both of these conjectures are settled under almost all of the given constraints.