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Semisymmetric elementary abelian covers of the Möbius-Kantor graphMalnič, Aleksander ...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 ▫$\pw_M$▫ 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. Malnic, D. Marušic, P. Potocnik, 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 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 ▫$p \equiv 5,9,13,17,21 \pmod{24}$▫ and ▫$p \equiv 1 \pmod{24}$▫, respectively, and to ▫$(p+1)/4$▫ and ▫$1+(p+1)/4$▫ in cases ▫$p \equiv 3,7,11,15,23 \pmod{24}$▫ and ▫$p \equiv 19 \pmod{24}$▫, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly.Source: Discrete mathematics. - ISSN 0012-365X (Vol. 307, iss. 17-18, 2007, str. 2156-2175)Type of material - article, component partPublish date - 2007Language - englishCOBISS.SI-ID - 14337113
Author
Malnič, Aleksander |
Marušič, Dragan |
Miklavič, Štefko |
Potočnik, Primož, 1971-
Topics
matematika |
teorija grafov |
graf |
krovna projekcija |
dvig avtomorfizmov |
homološka grupa |
matrična grupa |
invariantni podprostori |
mathematics |
graph theory |
graph |
covering projection |
lifting automorphisms |
homology group |
group representation |
matrix group |
invariant subspaces
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Database name | Field | Year |
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Malnič, Aleksander | 02507 |
Marušič, Dragan | 02887 |
Miklavič, Štefko | 21656 |
Potočnik, Primož, 1971- | 18838 |
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