Arc-transitive abelian regular covering graphs

Abstract

A lot of attention has been paid recently to the construction of symmetric covers of symmetric graphs. After a new approach given by Conder and the author [Arc-transitive abelian regular covers of cubic graphs, J. Algebra 387 (2013) 215–242], the group of covering transformations can be extended to more general abelian groups rather than cyclic or elementary abelian groups. In this paper, by using the Conder–Ma approach, we investigate the symmetric covers of 4-valent symmetric graphs. As an application, all the arc-transitive abelian regular covers of the 4-valent complete graph [Formula: see text] which can be obtained by lifting the arc-transitive subgroups of automorphisms [Formula: see text] and [Formula: see text] are classified.