Semisymmetric elementary abelian covers of the Möbius-Kantor graph
Semisymmetric elementary abelian covers of the Möbius-Kantor graph
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DOI：
10.1016/j.disc.2006.10.008
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发表时间：
2005-10
期刊：
影响因子：
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通讯作者：
A. Malnic;D. Marušič;Stefko Miklavic;P. Potočnik
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文献类型：
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作者：
A. Malnic;D. Marušič;Stefko Miklavic;P. Potočnik
设N：X →X是连通图的正则覆盖投影，其覆盖变换群同构于N.如果N是初等阿贝尔p-群，则投影N称为p-初等阿贝尔群。如果Aut X的某个点传递（边传递）子群沿着<$N提升，则投影<$N是点传递（边传递）的;如果它是边传递但不是点传递的，则投影<$N是半对称的。投影<$N是极小半对称的，如果<$N不能写成两个（非平凡的）正则覆盖投影的合成<$N=<$<$N <$M，其中<$M是半对称的。寻找初等阿贝尔覆盖投影可以通过作用于图的第一同调群的自同构的线性表示来组合地把握。该方法本质上简化为寻找素域上矩阵群的不变子空间（参见[A. Malniverse，D. Maruši，P. Potočnik，Elementary abelian covers of graphs，J. Algebras Combin. 20（2004）71-97]）。本文构造了Möbius-Kantor图--广义Petersen图GP（8，3）的所有两两非同构的极小半对称初等阿贝尔正则覆盖投影.对于p=2，不存在这样的覆盖。否则，在p = 5，9，13，17，21（mod 24）和p = 1（mod 24）的情况下，这种覆盖投影的数目分别等于（p-1）/4和1+（p-1）/4，而在p = 3，7，11，15，23（mod 24）和p = 19（mod 24）的情况下，这种覆盖投影的数目分别等于（p+1）/4和1+（p+1）/4。对于每个这样的覆盖投影，显式地显示生成对应覆盖的电压规则。
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 ℘Nis called p-elementary abelian. The projection ℘Nis 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 ℘Nis minimal semisymmetric if ℘Ncannot be written as a composition ℘N=℘∘℘Mof two (nontrivial) regular covering projections, where ℘Mis 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 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.