Semisymmetric elementary abelian covers of the Möbius-Kantor graph
| dc.creator | Malnic, A. | |
| dc.creator | Marusic, D. | |
| dc.creator | Miklavic, S. | |
| dc.creator | Potocnik, P. | |
| dc.date | 2005-10-18 | |
| dc.date.accessioned | 2026-07-07T06:47:37Z | |
| dc.date.available | 2026-07-07T06:47:37Z | |
| dc.description | Let $\p_N \colon \tX \to X$ 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 $\p_N$ is called $p$-elementary abelian. The projection $\p_N$ is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of $\Aut X$ lifts along $\p_N$, and semisymmetric if it is edge- but not vertex-transitive. The projection $\p_N$ is minimal semisymmetric if $p_N$ cannot be written as a composition $\p_N = \p \circ \p_M$ of two (nontrivial) regular covering projections, where $\p_M$ 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 {\em J. Algebr. Combin.}, {\bf 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 $\GP(8,3)$, 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 $p \equiv 5,9,13,17,21 (\mod 24)$ and $p \equiv 1 (\mod 24)$, respectively, and to $(p+1)/4$ and $1+ (p+1)/4$ in cases $p \equiv 3,7,11,15,23 (\mod 24)$ and $p \equiv 19 (\mod 24)$, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly. | |
| dc.description | 23 pages, 8 figures | |
| dc.identifier | https://arxiv.org/abs/math/0510383 | |
| dc.identifier | http://arxiv.org/abs/math/0510383 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/103712 | |
| dc.subject | Combinatorics | |
| dc.subject | 05C25; 05C70 | |
| dc.title | Semisymmetric elementary abelian covers of the Möbius-Kantor graph | |
| dc.type | text |