Let $G=(V,E)$ be a graph. For $e=uv\in E(G)$, $n_u(e)$ is the number of vertices of $G$ lying closer to $u$ than to $v$ and $n_v(e)$ is the number of vertices of $G$ lying closer to $v$ than $u$. The $GA_2$ index of $G$ is defined as $\sum_{uv \in E(G)}\frac{2\sqrt{n_u(e)n_v(e)}}{n_u(e)+n_v(e)}$. We explore here some mathematical properties and present explicit formulas for this new index under several graph operations.