On a class of vertex-primitive arc-transitive amply regular graphs

A simple $k$-regular graph with $v$ vertices is an amply regular graph with parameters $(v, k, \lambda, \mu)$ if any two adjacent vertices have exactly $\lambda$ common neighbors and any two vertices which are at distance $2$ in this graph have exactly $\mu$ common neighbors. Let $G$ be a finite group, $H \le G$, ${\mathfrak{H}} = \{H^g \,|\, g \in G \}$ be the corresponding conjugacy class of subgroups of $G$, and $1 \le d$ be an integer. We construct a simple graph $\Gamma(G, H, d)$ as follows. The vertices of $\Gamma(G, H, d)$ are the elements of ${\mathfrak{H}}$, and two vertices $H_1$ and $H_2$ from ${\mathfrak{H}}$ are adjacent in $\Gamma(G, H, d)$ if and only if $|H_1 \cap H_2| = d$. In this paper we prove that if $q$ is a prime power with $13 \le q \equiv 1 \pmod{4}$, $G=SL_2(q)$, and $H$ is a dihedral maximal subgroup of $G$ of order $2(q-1)$, then the graph $\Gamma(G, H, 8)$ is a vertex-primitive arc-transitive amply regular graph with parameters $\left(\dfrac{q(q+1)}{2}, \dfrac{q-1}{2}, 1, 1\right)$ and with ${\rm Aut}(PSL_2(q))\le {\rm Aut}(\Gamma)$. Moreover, we prove that $\Gamma(G, H, 8)$ has a perfect $1$-code, in particular, its diameter is more than $2$.