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 1 - 10 / 1212 1.Primitive groups of degree twice a prime numberDragan Marušič, 2012, unpublished invited conference lectureFound in: ključnih besedahSummary of found: ...primitive group, strongly regular graph, primitive action, ...Keywords: primitive group, strongly regular graph, primitive actionPublished: 15.10.2013; Views: 1871; Downloads: 24 Full text (0,00 KB) 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: ključnih besedahSummary of found: ...classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such...Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference setPublished: 15.10.2013; Views: 1451; Downloads: 66 Full text (0,00 KB) 3.On 2-fold covers of graphsYan-Quan Feng, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2008, original scientific articleAbstract: A regular covering projection ▫$\wp : \widetilde{X} \to X$▫ of connected graphs is ▫$G$▫-admissible if ▫$G$▫ lifts along ▫$\wp$▫. Denote by ▫$\tilde{G}$▫ the lifted group, and let CT▫$(\wp)$▫ be the group of covering transformations. The projection is called ▫$G$▫-split whenever the extension ▫{$\mathrm{CT}}(\wp) \to \tilde{G} \to G$▫ splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that ▫$G$▫ is transitive on ▫$X$▫, a ▫$G$▫-split cover is said to be ▫$G$▫-split-transitive if all complements ▫$\tilde{G} \cong G$▫ of CT▫$(\wp)$▫ within ▫$\tilde{G}$▫ are transitive on ▫$\widetilde{X}$▫; it is said to be ▫$G$▫-split-sectional whenever for each complement ▫$\tilde{G}$▫ there exists a ▫$\tilde{G}$▫-invariant section of ▫$\wp$▫; and it is called ▫$G$▫-split-mixed otherwise. It is shown, when ▫$G$▫ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. Split-transitive covers, however, are considerably more difficult to analyze. For cubic symmetric graphs split 2-cover are necessarily canonical double covers (that is, no ▫$G$▫-split-transitive 2-covers exist) when ▫$G$▫ is 1-regular or 4-regular. In all other cases, that is, if ▫$G$▫ is ▫$s$▫-regular, ▫$s=2,3$▫ or ▫$5$▫, a necessary and sufficient condition for the existence of a transitive complement ▫$\tilde{G}$▫ is given, and moreover, an infinite family of split-transitive 2-covers based on the alternating groups of the form ▫$A_{12k+10}$▫ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group ▫$G$▫ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed.Found in: ključnih besedahSummary of found: ...A regular covering projection ▫$\wp : \widetilde{X} \to X$▫... ...along ▫$\wp$▫. Denote by ▫$\tilde{G}$▫ the lifted group, and let CT▫$(\wp)$▫ be the group of...Keywords: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projectionPublished: 15.10.2013; Views: 1604; Downloads: 15 Full text (0,00 KB) 4.Hamilton cycles in (2, odd, 3)-Cayley graphsHenry Glover, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2012, original scientific articleAbstract: In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite ▫$(2, s, 3)$▫-generated group ▫$G = \langle a, x| a^2 = x^s = (ax)^3 = 1, \dots \rangle$▫ has a Hamilton path when ▫$|G|$▫ is congruent to 0 modulo 4, and has a Hamilton cycle when ▫$|G|$▫ is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from ▫$|G|$▫, also ▫$s$▫ is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever ▫$|G|$▫ is congruent to 0 modulo 4 and s is odd. This leaves ▫$|G|$▫ congruent to 0 modulo 4 with s congruent to 2 modulo 4 as the only remaining open case. In this last case, however, the "tree of faces" approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in cubic Cayley graphs arising from finite ▫$(2, s, 3)$▫-generated groups.Found in: ključnih besedahSummary of found: ...Cayley graph, Hamilton cycle, arc-transitive graph, 1- regular action, automorphism group...Keywords: Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism groupPublished: 15.10.2013; Views: 1393; Downloads: 70 Full text (0,00 KB) 5.On overgroups of regular abelian p-groupsEdward Dobson, 2009, original scientific articleAbstract: Let ▫$G$▫ be a transitive group of odd prime-power degree whose Sylow ▫$p$▫-subgroup ▫$P$▫ is abelian od rank ▫$t$▫. Weshow that if ▫$p > 2^{t-1}$▫, then ▫$G$▫ has a normal subgroup that is a direct product of ▫$t$▫ permutation groups of smaller degree that are either cyclic or doubly-transitive simple groups. As a consequence, we determine the full automorphism group of a Cayley diagraph of an abelian group with rank two such that the Sylow ▫$p$▫-subgroup of the full automorphism group is abelian.Found in: ključnih besedahSummary of found: ...Let ▫$G$▫ be a transitive group of odd prime-power degree whose Sylow ▫$p$▫-subgroup...Keywords: group theory, graph theory, Cayley graph, abelian group, regular group, p-groupPublished: 15.10.2013; Views: 1786; Downloads: 80 Full text (0,00 KB) 6.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: ključnih besedahSummary of found: ...classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such...Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference setPublished: 10.07.2015; Views: 1009; Downloads: 54 Full text (0,00 KB) 7.Algebraični aspekti teorije grafovAdemir Hujdurović, 2013, doctoral dissertationFound in: ključnih besedahSummary of found: ...circulant, bicirculant, semi regular automorphism, vertex-transitive graph, half-arc-transitive graph, snark, Cayley... ...action, regular cover of a graph, automorphism group, ...Keywords: circulant, bicirculant, semiregular automorphism, vertex-transitive graph, half-arc-transitive graph, snark, Cayley graph, quasi m-Cayley graph, generalized Cayley graph, I-regular action, regular cover of a graph, automorphism groupPublished: 10.07.2015; Views: 1773; Downloads: 8 Full text (0,00 KB) 8.Classification of regular maps of Euler characteristic -3pMarston D. E. Conder, Roman Nedela, Jozef Širan, 2012, original scientific articleFound in: ključnih besedahSummary of found: ...grupa avtomorfizmov, Eulerjeva karakteristika, regular map, automorphism group, Euler characteristics, ...Keywords: regularen zemljevid, grupa avtomorfizmov, Eulerjeva karakteristika, regular map, automorphism group, Euler characteristicsPublished: 15.10.2015; Views: 1352; Downloads: 82 Full text (0,00 KB) 9.Computing stable epimorphisms onto finite groupsRok Požar, 2018, original scientific articleFound in: ključnih besedahSummary of found: ...algorith, epimorphism, finitely presented group, regular covering projection, quotient group...Keywords: algorith, epimorphism, finitely presented group, regular covering projection, quotient groupPublished: 02.03.2018; Views: 738; Downloads: 95 Full text (0,00 KB) 10.On even-closed regular embeddings of graphsIstván Kovács, Klavdija Kutnar, Dragan Marušič, Daniel Pellicer, 2018, original scientific articleFound in: ključnih besedahSummary of found: ...odd action automorphism, even action automorphism, automorphism group, vertex-transitive graph, orientable-regular map...Keywords: odd action automorphism, even action automorphism, automorphism group, vertex-transitive graph, orientable-regular mapPublished: 18.07.2018; Views: 1194; Downloads: 98 Full text (0,00 KB)
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