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 d4a2; can be expressed as R ( G ) = ∑ x y ∈ Y ( G ) 1 τ ( 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 d4b4;(d4a2;) 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.