| Title: | Semisymmetric elementary abelian covers of the Möbius-Kantor graph |
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| Authors: | ID Malnič, Aleksander (Author)
ID Marušič, Dragan (Author)
ID Miklavič, Štefko (Author)
ID Potočnik, Primož (Author) |
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| Files: |  http://dx.doi.org/10.1016/j.disc.2006.10.008
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| Language: | English |
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| Work type: | Not categorized |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | IAM - Andrej Marušič Institute
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| Abstract: | Let ℘N:˜X→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 ℘N is called p-elementary abelian. The projection ℘N is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of Aut X lifts along ℘N, and semisymmetric if it is edge- but not vertex-transitive. The projection ℘N is minimal semisymmetric if ℘N cannot be written as a composition ℘N=℘∘℘M of two (nontrivial) regular covering projections, where \pwM 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≡5,9,13,17,21(mod24) and p≡1(mod24), respectively, and to (p+1)/4 and 1+(p+1)/4 in cases p≡3,7,11,15,23(mod24) and p≡19(mod24), respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly. |
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| Keywords: | mathematics, graph theory, graph, covering projection, lifting automorphisms, homology group, group representation, matrix group, invariant subspaces |
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| Year of publishing: | 2007 |
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| Number of pages: | str. 2156-2175 |
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| Numbering: | Vol. 307, iss. 17-18 |
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| PID: | 20.500.12556/RUP-7723  |
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| ISSN: | 0012-365X |
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| UDC: | 519.17 |
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| COBISS.SI-ID: | 14337113 |
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| Publication date in RUP: | 03.04.2017 |
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| Views: | 3003 |
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| Downloads: | 92 |
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MALNIČ, Aleksander, MARUŠIČ, Dragan, MIKLAVIČ, Štefko and POTOČNIK, Primož, 2007, Semisymmetric elementary abelian covers of the Möbius-Kantor graph. [online]. 2007. Vol. 307, no. 17–18, p. 2156–2175. [Accessed 4 April 2025]. Retrieved from: http://dx.doi.org/10.1016/j.disc.2006.10.008
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