On 2-arc-transitive graphs of product action type

In this paper, we discuss the structural information about 2-arc-transitive (non-bipartite and bipartite) graphs of product action type. It is proved that a 2-arc-transitive graph of product action type requires certain restrictions on either the vertex-stabilizers or the valency. Based on the existence of some equidistant linear codes, a construction is given for 2-arc-transitive graphs of non-diagonal product action type, which produces several families of such graphs. Besides, a nontrivial construction is given for 2-arc-transitive bipartite graphs of diagonal product action typeComment: 18 page

Intersection density of imprimitive groups of degree pqpq

A subset $\mathcal{F}$ of a finite transitive group $G\leq \operatorname{Sym}(\Omega)$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on an element of $\Omega$. The \emph{intersection density} of $G$ is the number \rho(G) = \max\left\{ \frac{\mathcal{|F|}}{|G|/|\Omega|} \mid \mathcal{F}\subset G \mbox{ is intersecting} \right\}. Recently, Hujdurovi\'{c} et al. [Finite Fields Appl., 78 (2022), 101975] disproved a conjecture of Meagher et al. (Conjecture~6.6~(3) in [ J.~Combin. Theory, Ser. A 180 (2021), 105390]) by constructing equidistant cyclic codes which yield transitive groups of degree $pq$, where $p = \frac{q^k-1}{q-1}$ and $q$ are odd primes, and whose intersection density equal to $q$. In this paper, we use the cyclic codes given by Hujdurovi\'{c} et al. and their permutation automorphisms to construct a family of transitive groups $G$ of degree $pq$ with $\rho(G) = \frac{q}{k}$, whenever $k<q<p=\frac{q^k-1}{q-1}$ are odd primes. Moreover, we extend their construction using cyclic codes of higher dimension to obtain a new family of transitive groups of degree a product of two odd primes $q<p = \frac{q^k-1}{q-1}$, and whose intersection density are equal to $q$. Finally, we prove that if $G\leq \operatorname{Sym}(\Omega)$ of degree a product of two arbitrary odd primes $p>q$ and $\left\langle \bigcup_{\omega\in \Omega} G_\omega \right\rangle$ is a proper subgroup, then $\rho(G) \in \{1,q\}$