A characterisation of edge-affine $2$-arc-transitive covers of $\K_{2^n,2^n}$

Abstract

We introduce the notion of an \emph{$n$-dimensional mixed dihedral group}, a general class of groups for which we give a graph theoretic characterisation. In particular, if $H$ is an $n$-dimensional mixed dihedral group then the we construct an edge-transitive Cayley graph $\Gamma$ of $H$ such that the clique graph $\Sigma$ of $\Gamma$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(\Sigma)$ inducing a particular \emph{edge-affine} action on $\K_{2^n,2^n}$. Conversely, we prove that if $\Sigma$ is a $2$-arc-transitive normal cover of $\K_{2^n,2^n}$, with a subgroup of $\Aut(\Sigma)$ inducing an \emph{edge-affine} action on $\K_{2^n,2^n}$, then the line graph $\Gamma$ of $\Sigma$ is a Cayley graph of an $n$-dimensional mixed dihedral group. Furthermore, we give an explicit construction of a family of $n$-dimensional mixed dihedral groups. This family addresses a problem proposed by Li concerning normal covers of prime power order of the `basic' $2$-arc-transitive graphs. In particular, we construct, for each $n\geq 2$, a $2$-arc-transitive normal cover of $2$-power order of the `basic' graph $\K_{2^n,2^n}$.