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

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-2n,K-2n for n >= 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 2n, such that X boolean AND Y = 1, H is generated by X boolean OR Y, and H/H ' congruent to X x Y. Our characterisation shows that each 2-arc-transitive normal cover of K-2n,K-2n 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-2n,K-2n induces an edge-affine action on K-2n,K-2n (and we show that such actions are one of just four possible types of 2-arc-transitive actions on K-2n,K-2n). As a partial converse, we provide a graph theoretic characterisation of n-dimensional mixed dihedral groups, and finally, for each n >= 2, we give an explicit construction of an n-dimensional mixed dihedral group which is a 2-group of order 2(n2+2n), and a corresponding 2-arc-transitive normal cover of 2-power order of K-2n,K-2n. 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. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and datamining, AI training, and similar technologies.