A characterisation of edge-affine 2-arc-transitive covers of K2n,2n

Abstract

The family of finite 2-arc-transitive graphs of a given valency is closed under forming non-trivial normal quotients, and graphs in this family having no non-trivial normal quotient are called ‘basic’. To date, the vast majority of work in the literature has focused on classifying these ‘basic’ graphs. By contrast we give here a characterisation of the normal covers of the ‘basic’ 2-arc-transitive graphs $K_{2^n,2^n}$ for $n\ge 2$. The characterisation identified the special role of graphs associated with a subgroup of automorphisms called an n-dimensional mixed dihedral group. This is a group H with two subgroups X and Y, each elementary abelian of order $2^n$, such that $X\cap Y=1$, H is generated by $X\cup Y$, and $H/H^{\prime }\cong X\times Y$.

Our characterisation shows that each 2-arc-transitive normal cover of $K_{2^n,2^n}$ is either itself a Cayley graph, or is the line graph of a Cayley graph of an n-dimensional mixed dihedral group. In the latter case, we show that the 2-arc-transitive group acting on the normal cover of $K_{2^n,2^n}$ induces an edge-affine action on $K_{2^n,2^n}$ (and we show that such actions are one of just four possible types of 2-arc-transitive actions on $K_{2^n,2^n}$). As a partial converse, we provide a graph theoretic characterisation of n-dimensional mixed dihedral groups, and finally, for each $n\ge 2$, we give an explicit construction of an n-dimensional mixed dihedral group which is a 2-group of order $2^{n^2+2n}$, and a corresponding 2-arc-transitive normal cover of 2-power order of $K_{2^n,2^n}$. Note that these results partially address a problem proposed by Caiheng Li concerning normal covers of prime power order of the ‘basic’ 2-arc-transitive graphs.