Let $G$ be a finite group, $H$ is proper subgroup and ${\cal H}(H)$ the set of all subgroups of conjugate H. Construct the graph $\Gamma$, with vertices ${\cal H}(H)$ and two subgroups adjacent if they have trival intersection.
Its true (if graph not trivial), that this graph regular and have diamert 2?
For example in case $G = A_5$, $H = S_3$ (generated by $(12)(34)$ and $(125)$), $\Gamma$ is complement of Petersen graph. For other $H$, $\Gamma$ is complete multipartite graph (complenet of $mK_n$ for some $n$ and $m$) (for $H \in \{C_2, C_3, C_2\times C_2, C_5\}$) and empty graph (for $H \in \{D_5, A_4\}$)
Next interest example gave $G = A_6$, and ${\cal H}(H) = Syl_2(A_6)$. Then a $\Gamma$ has order $45$, degree $32$, spectrum $\{32^1, 2^{19}, -1^{16}, -6^9\}$, $\Gamma$ is vertex transitive and has automorphism group isomorphic to $(A_{6} \rtimes C_{2}) \rtimes C_{2}$.