Extremal results for the hyperbolic Sombor index of molecular trees

For a graph G , the hyperbolic Sombor index is defined as H S O ( G ) = ∑ u v ∈ E d u 2 + d v 2 min d u , d v . Where d u and d v denote the degrees of the vertices u and v , respectively. This recently introduced index has attracted considerable attention due to its sensitivity to graph structure and its potential applications in chemical graph theory. In this paper, the extremal bounds for the hyperbolic Sombor index of molecular trees in terms of the order are determined. Furthermore, the bounds for the hyperbolic Sombor index of molecular trees in terms of the order and number of pendent vertices are determined. Moreover, the trees attaining these bounds are characterized.