Sharp bounds of the Randić index for trees with a given metric dimension

The Randić index, a significant topological index in chemical graph theory, is widely used to model molecular stability and reactivity. For a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], the Randić index is defined as [Formula: see text] where [Formula: see text] and [Formula: see text] are the degrees of vertices [Formula: see text] and [Formula: see text]. In this study, we focus on trees and establish bounds on the Randić index in terms of both the order and the metric dimension, a parameter that reflects the uniqueness of vertex identification by distances to a set of reference points. This mathematical model provides chemists with an effective, non-empirical method for predicting molecular properties based on structural parameters alone. Our findings highlight the extremal structures of trees that achieve these bounds, offering new insights into the application of the Randić index for analyzing molecular structures and properties in chemical graph theory.