On the Hyperbolic Sombor Index of Graphs

The hyperbolic Sombor index (HSO) is a recently introduced vertex-degree–based topological index that originates from the geometric properties of a hyperbola. In this work, we explore several mathematical properties of the HSO index, as well as revisit and refine some previously reported results. We first provide a counterexample to the claim that HSO(G) always increases with the addition of an edge and establish a sufficient condition under which this monotonicity holds. We then present refined versions of some existing results and proofs. Furthermore, we establish sharp upper and lower bounds for the HSO index across various classes of graphs, including trees, unicyclic graphs, and bicyclic graphs, and characterize the corresponding extremal graphs that attain these bounds. Finally, we identify the first eight minimal trees, as well as seven minimal unicyclic and bicyclic graphs with respect to HSO.