91. On 2-fold covers of graphsYan-Quan Feng, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2008, original scientific article Abstract: 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.
Keywords: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection
Published in RUP: 15.10.2013; Views: 3687; Downloads: 34
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92. Decomposition of skew-morphisms of cyclic groupsIstván Kovács, Roman Nedela, 2011, original scientific article Abstract: A skew-morphism of a group ▫$H$▫ is a permutation ▫$\sigma$▫ of its elements fixing the identity such that for every ▫$x, y \in H$▫ there exists an integer ▫$k$▫ such that ▫$\sigma (xy) = \sigma (x)\sigma k(y)$▫. It follows that group automorphisms are particular skew-morphisms. Skew-morphisms appear naturally in investigations of maps on surfaces with high degree of symmetry, namely, they are closely related to regular Cayley maps and to regular embeddings of the complete bipartite graphs. The aim of this paper is to investigate skew-morphisms of cyclic groups in the context of the associated Schur rings. We prove the following decomposition theorem about skew-morphisms of cyclic groups ▫$\mathbb Z_n$▫: if ▫$n = n_{1}n_{2}$▫ such that ▫$(n_{1}n_{2}) = 1$▫, and ▫$(n_{1}, \varphi (n_{2})) = (\varphi (n_{1}), n_{2}) = 1$▫ (▫$\varphi$▫ denotes Euler's function) then all skew-morphisms ▫$\sigma$▫ of ▫$\mathbb Z_n$▫ are obtained as ▫$\sigma = \sigma_1 \times \sigma_2$▫, where ▫$\sigma_i$▫ are skew-morphisms of ▫$\mathbb Z_{n_i}, \; i = 1, 2$▫. As a consequence we obtain the following result: All skew-morphisms of ▫$\mathbb Z_n$▫ are automorphisms of ▫$\mathbb Z_n$▫ if and only if ▫$n = 4$▫ or ▫$(n, \varphi(n)) = 1$▫.
Keywords: cyclic group, permutation group, skew-morphism, Schur ring
Published in RUP: 15.10.2013; Views: 3932; Downloads: 109
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93. Distance-regular Cayley graphs on dihedral groupsŠtefko Miklavič, Primož Potočnik, 2007, original scientific article Abstract: 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.
Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set
Published in RUP: 15.10.2013; Views: 3094; Downloads: 99
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94. Semiovals contained in the union of three concurrent linesAart Blokhuis, György Kiss, István Kovács, Aleksander Malnič, Dragan Marušič, János Ruff, 2007, original scientific article Abstract: Semiovals which are contained in the union of three concurrent lines are studied. The notion of a strong semioval is introduced, and a complete classification of these objects in PG▫$(2,p)$▫ and PG▫$(2,p^2)$▫, ▫$p$▫ an odd prime, is given.
Keywords: mathematics, semioval, group factorization
Published in RUP: 15.10.2013; Views: 2930; Downloads: 130
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98. On quartic half-arc-transitive metacirculantsDragan Marušič, Primož Šparl, 2008, original scientific article Abstract: Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ▫$\rho$▫ and ▫$\sigma$▫, where ▫$\rho$▫ is ▫$(m,n)$▫-semiregular for some integers ▫$m \ge 1$▫, ▫$n \ge 2▫$, and where ▫$\sigma$▫ normalizes ▫$\rho$▫, cyclically permuting the orbits of ▫$\rho$▫ in such a way that ▫$\sigma^m$▫ has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.
Keywords: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group
Published in RUP: 15.10.2013; Views: 3849; Downloads: 132
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100. Classification of 2-arc-transitive dihedrantsShao Fei Du, Aleksander Malnič, Dragan Marušič, 2008, original scientific article Abstract: A complete classification of 2-arc-transitive dihedrants, that is, Cayley graphs of dihedral groups is given, thus completing the study of these graphs initiated by the third author in [D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162-196]. The list consists of the following graphs: (i) cycles ▫$C_{2n},\; n \ge 3$▫; (ii) complete graphs ▫$K_{2n}, \; n \ge 3$▫; (iii) complete bipartite graphs ▫$K_{n,n}, \; n \ge 3$▫; (iv) complete bipartite graphs minus a matching ▫$K_{n,n} - nK_2, \; n \ge 3$▫; (v) incidence and nonincidence graphs ▫$B(H_{11})$▫ and ▫$B'(H_{11})$▫ of the Hadamard design on 11 points; (vi) incidence and nonincidence graphs ▫$B(PG(d,q))$▫ and ▫$B'(PG(d,q))$▫, with ▫$d \ge 2$▫ and ▫$q$▫ a prime power, of projective spaces; (vii) and an infinite family of regular ▫${\mathbb{Z}}_d$▫-covers ▫$K_{q+1}^{2d}$▫ of ▫$K_{q+1, q+1} - (q+1)K_2$▫, where ▫$q \ge 3$▫ is an odd prime power and ▫$d$▫ is a divisor of ▫$\frac{q-1}{2}$▫ and ▫$q-1$▫, respectively, depending on whether ▫$q \equiv 1 \pmod{4}$▫ or ▫$q \equiv 3 \pmod{4}$▫ obtained by identifying the vertex set of the base graph with two copies of the projective line ▫$PG(1,q)$▫, where the missing matching consists of all pairs of the form ▫$[i,i']$▫, ▫$i \in PG(1,q)$▫, and the edge ▫$[i,j']$▫ carries trivial voltage if ▫$i=\infty$▫ or ▫$j=\infty$▫, and carries voltage ▫$\bar{h} \in {\mathbb{Z}}_d$▫, the residue class of ▫$h \in {\mathbb{Z}}_d$▫, if and only if ▫$i-j = \theta^h$▫, where ▫$\theta$▫ generates the multiplicative group ▫${\mathbb{F}}_q^\ast$▫ of the Galois field ▫${\mathbb{F}}_q$▫.
Keywords: permutation group, imprimitive group, dihedral group, Cayley graph, dihedrant, 2-Arc-transitive graph
Published in RUP: 15.10.2013; Views: 3469; Downloads: 89
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