Semisymmetric elementary abelian covers of the Möbius–Kantor graph Original Research Article

Let image be a regular covering projection of connected graphs with the group of covering transformations isomorphic to N. If N is an elementary abelian p-group, then the projection image is called p-elementary abelian. The projection image is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of Aut image lifts along image, and semisymmetric if it is edge- but not vertex-transitive. The projection image is minimal semisymmetric if image cannot be written as a composition image of two (nontrivial) regular covering projections, where image is semisymmetric. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields (see [A. Malnič, D. Marušič, P. Potočnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004) 71–97]). In this paper, all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius–Kantor graph, the Generalized Petersen graph image, are constructed. No such covers exist for image. Otherwise, the number of such covering projections is equal to image and image in cases image and image, respectively, and to image and image in cases image and image, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly.