On the diameter of intersection graphs of finite groups

The intersection graph ${\mathrm{\Delta }}_G$ of a finite group $G$ is a simple graph with vertices the non-trivial proper subgroups of $G$, and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were introduced by Csákány and Pollák in 1969. In this paper we answer two long-standing open questions posed by Csákány and Pollák concerning the diameter of intersection graphs. We prove some necessary conditions for a non-simple group to have an intersection graph of diameter 4. We also construct the first examples of non-simple groups and alternating groups whose intersection graphs have diameter 4.