New Sharp Extremal Bounds for the Randić Index of Trees With Prescribed Roman Domination Number

The Randić index is a classical degree-based topological index that captures branching features of a graph and has broad applications in chemical graph theory and related network models. Roman domination is a defense-inspired covering concept in which vertices are assigned protective labels so that every unprotected vertex is adjacent to a strongly protected one, and the Roman domination number measures the minimum total defensive weight needed. In this paper, we investigate how these two invariants interact on trees. We establish new sharp lower bounds for the Randić index of a tree expressed in terms of its order and its Roman domination number, and we characterize all trees attaining equality. We further derive a matching upper bound in the same parameters and again determine the extremal structures. In particular, we show that among trees with fixed order and Roman domination number, the star uniquely attains the minimum Randić index, while the path uniquely attains the maximum. A comparative analysis with previously known bounds demonstrates that the new lower bound is consistently tighter and tracks the exact Randić index more closely, especially for larger trees. We conclude with an open problem on extending these extremal results to unicyclic and bicyclic graphs.