The intersection density of cubic arc-transitive graphs with $2$-arc-regular full automorphism group equal to $\operatorname{PGL}_2(q)$
Karen Meagher, Andriaherimanana Sarobidy Razafimahatratra
TL;DR
This work determines the weak intersection density array for cubic arc-transitive graphs of type {1,2^1} with full automorphism group $\operatorname{PGL}_2(q)$ and a $1$-arc-regular subgroup $\operatorname{PSL}_2(q)$, resolving open cases when $q \equiv 2 \pmod{3}$ and detailing densities across congruence classes mod $3$ and mod $5$. The approach combines orbital/derangement graph techniques, suborbit analysis, and explicit normalizer/transversal structures in $\operatorname{PSL}_2(q)$ and $\operatorname{PGL}_2(q)$, yielding precise descriptions of the first subconstituents as unions of orbital graphs or Cayley graphs on cyclic groups. The main results enumerate the possible weak intersection density arrays $\overline{\rho}(X)$ by case: for $q=3^k$ with $k$ odd or even, and for $q \equiv 1\pmod{3}$ or $q \equiv 2\pmod{3}$ with further mod $5$ conditions, with explicit densities such as $[3^{k-1}]$, $[3^{\frac{k}{2}-1}]$, $[1,2]$, $[1,4/3]$, or $[1]$ in the respective regimes. The paper also proves that the $2$-arc-regular group $\operatorname{PGL}_2(q)$ attains density $\rho(G^*)=1$ in all considered cases, completing the density classification for the family and contributing to the broader understanding of intersection densities in transitive permutation groups.
Abstract
The \emph{intersection density} of a transitive permutation group $G\leq \operatorname{Sym}(Ω)$ is the ratio between the largest size of a subset of $G$ in which any two agree on at least one element of $Ω$, 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 $G^* = \operatorname{PGL}_2(q)$ and an arc-regular subgroup of automorphism $G = \operatorname{PSL}_2(q)$.