The variation of Randi\'c index $R'(G)$ of a graph G is defined by $R'(G)=\sum_{uv}\frac1{\max\{d_u,d_v\}}$, where $d_u$ is the degree of a vertex $u$ in $G$ and the summation extends over all edges $uv$ of $G$. In this work, we characterize the extremal trees achieving the minimum value of $R'$ for trees with given number of vertices and leaves. Furthermore, we characterize the extremal graphs achieving the minimum value of $R'$ for connected graphs with given number of vertices and girth.