Bounds for the Hyperbolic Sombor Index in Terms of the Variable Euler–Sombor Index and Relations to Other Degree-Based Invariants

The Hyperbolic Sombor index HSO(G) of a graph G=(V(G),E(G)) is proposed as a new vertex-degree-based topological index. In this paper, we investigate the mathematical properties of this novel vertex-degree-based topological index for general graphs. We first establish tight upper and lower bounds on HSO(G) in terms of the variable Euler–Sombor index EU(λ,G) for all λ≥−2. Furthermore, we derive new bounds connecting HSO(G) with three other well-known degree-based invariants: the second Zagreb index M2(G), the Elliptic Sombor index ESO(G), and the Forgotten Sombor index FSO(G).