Sharp Lower Bound for Randić index of Trees with Fixed Roman Domination Number

Let \(G=(V,E)\) be a simple connected graph with vertex set \(V\) and edge set \(E\). The Randić index of graph \(G\) is the value \(R(G)=\sum_{uv\in E(G)} \frac{1}{\sqrt{d(u)d(v)}}\), where \(d(u)\) and \(d(v)\) refer to the degree of the vertices \(u\) and \(v\). We obtain a lower bound for the Randić index of trees in terms of the order and the Roman domination number, and we characterize the extremal trees for this bound.