The intersection density of cubic arc-transitive graphs with $2$-arc-regular full automorphism group equal to ${\mathrm{PGL}}_2(q)$

The \emph{intersection density} of a transitive permutation group $G\le \mathrm{Sym}(\mathrm{\Omega })$ is the ratio between the largest size of a subset of $G$ in which any two agree on at least one element of $\mathrm{\Omega }$, and the order of a point-stabilizer of $G$. In this paper, we determine the intersection densities of the automorphism group of the arc-transitive graphs admitting a $2$-arc-regular full automorphism group $G^{\ast }={\mathrm{PGL}}_2(q)$ and an arc-regular subgroup of automorphism $G={\mathrm{PSL}}_2(q)$.