Co-intersection graphs of nonabelian finite simple groups have diameter two

Let $G$ be a finite group. The co-intersection graph ${\mathrm{\Delta }}_G^c$ of $G$ has as its vertices the nontrivial proper subgroups of $G$, with edges joining those pairs of subgroups which intersect trivially. It is clear that every connected component of ${\mathrm{\Delta }}_G^c$ has diameter at most three. In this Note, we show that when $T$ is a nonabelian finite simple group, ${\mathrm{\Delta }}_T$ is connected of diameter exactly two. We also make several observations about the extent to which a finite group is determined by its co-intersection graph, and about the class of finite graphs arising as co-intersection graphs.