### On the intersection graph of a finite group

For a finite group $G$, the intersection graph of $G$ which is denoted by $\Gamma \left(G\right)$ is an undirected graph such that its vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H\cap K\ne 1$. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of $\mathrm{Aut}\left(\Gamma \right(G\left)\right)$.