Some Results on Hyperbolic Sombor Index

Let G(V,E) be a simple connected graph of order n and size m. The Hyperbolic Sombor index HSO(G)= \(\sum_{u v \in E(G)} \frac{\sqrt{d_u^2+d_v^2}}{\min \left\{d_u, d_v\right\}}\), where du and dv denote the degrees of vertices u and v, respectively, is a recently introduced degree-based topological index with growing significance in chemical graph theory. In this work, we derive sharpened and improved lower bounds for HSO(G), thereby refining the classical estimate HSO(G) ≥ m √2. Furthermore, we establish several new bounds for HSO(G) expressed in terms of important graph parameters such as the chromatic number, independence number, domination number and the mean and standard deviation of vertex degrees.