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 1 - 10 / 1112 1.Consistent Cycles in 1/2-Arc-Transitive GraphsMarko Boben, Štefko Miklavič, Primož Potočnik, 2009, original scientific articleFound in: osebiKeywords: mathematics, graph theory, 1/2-arc-transitivity, consistent cyclePublished: 15.10.2013; Views: 2380; Downloads: 8 Full text (0,00 KB)This document has more files! More... 2.Distance-regular Cayley graphs on dihedral groupsŠtefko Miklavič, Primož Potočnik, 2007, original scientific articleAbstract: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices.Found in: osebiKeywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference setPublished: 15.10.2013; Views: 1445; Downloads: 64 Full text (0,00 KB) 3.Rotary polygons in configurationsMarko Boben, Štefko Miklavič, Primož Potočnik, 2011, original scientific articleFound in: osebiKeywords: konfiguracija, poligon, avtomorfizem, antiavtomorfizemPublished: 15.10.2013; Views: 1577; Downloads: 33 Full text (0,00 KB)This document has more files! More... 4.Arc-transitive cycle decompositions of tetravalent graphsŠtefko Miklavič, Primož Potočnik, Stephen Wilson, 2008, original scientific articleAbstract: A cycle decomposition of a graph ▫$\Gamma$▫ is a set ▫$\mathcal{C}$▫ of cycles of ▫$\Gamma$▫ such that every edge of ▫$\Gamma$▫ belongs to exactly one cycle in ▫$\mathcal{C}$▫. Such a decomposition is called arc-transitive if the group of automorphisms of ▫$\Gamma$▫ that preserve setwise acts transitively on the arcs of ▫$\Gamma$▫. In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining and enumerating arc-transitive cycle decompositions admitted by a given arc-transitive tetravalent graph. Among other results we show that a connected tetravalent arc-transitive graph is either 2-arc-transitive, or is isomorphic to the medial graph of a reflexible map, or admits exactly one cycle structure.Found in: osebiKeywords: mathematics, graph theory, cycle decomposition, automorphism group, consistent cycle, medial mapsPublished: 15.10.2013; Views: 1605; Downloads: 55 Full text (0,00 KB) 5.Distance-regular Cayley graphs on dihedral groupsPrimož Potočnik, Štefko Miklavič, 2005, original scientific articleAbstract: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices.Found in: osebiKeywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference setPublished: 10.07.2015; Views: 1001; Downloads: 53 Full text (0,00 KB) 6.Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphsGabriel Verret, Primož Potočnik, Pablo Spiga, 2015, original scientific articleFound in: osebiKeywords: valenca 3, valenca 4, točkovna tranzitivnost, ločna tranzitivnost, lokalno-diedrski, valency 3, valency 4, vertex-transitive, arc-transitive, locally-dihedralPublished: 15.10.2015; Views: 1496; Downloads: 126 Full text (0,00 KB) 7.Semisymmetric elementary abelian covers of the Möbius-Kantor graphAleksander Malnič, Štefko Miklavič, Primož Potočnik, Dragan Marušič, 2007, original scientific articleAbstract: 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.Found in: osebiKeywords: mathematics, graph theory, graph, covering projection, lifting automorphisms, homology group, group representation, matrix group, invariant subspacesPublished: 03.04.2017; Views: 923; Downloads: 55 Full text (0,00 KB) 8.Two-arc-transitive two-valent digraphs of certain ordersKatja Berčič, Primož Potočnik, 2016, original scientific articleFound in: osebiKeywords: graph, digraph, arc-transitive, orderPublished: 08.08.2016; Views: 1081; Downloads: 71 Full text (0,00 KB) 9.Arc-transitive digraphs of given out-valency and with blocks of given sizePrimož Potočnik, Luke Morgan, Gabriel Verret, 2019, original scientific articleAbstract: Given integers ▫$k$▫ and ▫$m$▫, we construct a ▫$G$▫-arc-transitive graph of valency ▫$k$▫ and an ▫$L$▫-arc-transitive oriented digraph of out-valency ▫$k$▫ such that ▫$G$▫ and ▫$L$▫ both admit blocks of imprimitivity of size ▫$m$▫.Found in: osebiKeywords: arc-transitive digraphs, Cayley digraphs, imprimitive digraphsPublished: 28.06.2019; Views: 420; Downloads: 94 Full text (0,00 KB) 10.On reflexible polynomialsAleksander Malnič, Boštjan Kuzman, Primož Potočnik, 2018, published scientific conference contribution abstract (invited lecture)Found in: osebiKeywords: minimal elementary abelian cover, doubled cycle, polynomialPublished: 07.02.2018; Views: 1417; Downloads: 13 Full text (0,00 KB)
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