Extremal trees for the Randić index

Graph theory has applications in various fields due to offering important tools such as topological indices. Among the topological indices, the Randić index is simple and of great importance. The Randić index of a graph 𝒢 can be expressed as R(G)=∑xy∈Y(G)1τ(x)τ(y) R\left( G \right) = \sum\nolimits_{xy \in Y\left( G \right)} {{1 \over {\sqrt {\tau \left( x \right)\tau \left( y \right)} }}} , where 𝒴(𝒢) represents the edge set and τ(x) is the degree of vertex x. In this paper, considering the importance of the Randić index and applications two-trees graphs, we determine the first two minimums among the two-trees graphs.