The sum-connectivity index of a simple graph $G$ is defined in mathematical chemistry as $R^+(G)=\sum_{uv\in E(G)}=(d_u+d_v)^{-1/2}$, where $E(G)$ is the edge set of $G$ and $d_u$ is the degree of vertex $u$ in $G$. We give a best possible lower bound for the sum-connectivity index of a graph (a triangle-free graph, respectively) with $n$ vertices and minimum degree at least two and characterize the extremal graphs, where $n\geq11$.