Semisymmetric elementary abelian covers of the Möbius–Kantor graph

Abstract

Let ${\wp }_N:\tilde{X}\to X$ 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 ${\wp }_N$ is called p-elementary abelian. The projection ${\wp }_N$ is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of Aut $X$ lifts along ${\wp }_N$, and semisymmetric if it is edge- but not vertex-transitive. The projection ${\wp }_N$ is minimal semisymmetric if ${\wp }_N$ cannot be written as a composition ${\wp }_N=\wp \circ {\wp }_M$ of two (nontrivial) regular covering projections, where ${\wp }_M$ 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 $\mathrm{GP}(8,3)$, are constructed. No such covers exist for $p=2$. Otherwise, the number of such covering projections is equal to $(p-1)/4$ and $1+(p-1)/4$ in cases $p\equiv 5,9,13,17,21(\mathrm{mod}24)$ and $p\equiv 1(\mathrm{mod}24)$, respectively, and to $(p+1)/4$ and $1+(p+1)/4$ in cases $p\equiv 3,7,11,15,23(\mathrm{mod}24)$ and $p\equiv 19(\mathrm{mod}24)$, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly.