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1.
Vertex-transitive expansions of (1, 3)-trees
Marko Lovrečič Saražin, Dragan Marušič, 2010, published scientific conference contribution

Abstract: A nonidentity automorphism of a graph is said to be semiregular if all of its orbits are of the same length. Given a graph ▫$X$▫ with a semiregular automorphism ▫$\gamma$▫, the quotient of ▫$X$▫ relative to ▫$\gamma$▫ is the multigraph ▫$X/\gamma$▫ whose vertices are the orbits of ▫$\gamma$▫ and two vertices are adjacent by an edge with multiplicity ▫$r$▫ if every vertex of one orbit is adjacent to ▫$r$▫ vertices of the other orbit. We say that ▫$X$▫ is an expansion of ▫$X/\gamma$▫. In [J.D. Horton, I.Z. Bouwer, Symmetric ▫$Y$▫-graphs and ▫$H$▫-graphs, J. Combin. Theory Ser. B 53 (1991) 114-129], Hortonand Bouwer considered a restricted sort of expansions (which we will call :strong" in this paper) where every leaf of ▫$X/\gamma$▫ expands to a single cycle in ▫$X$▫. They determined all cubic arc-transitive strong expansions of simple ▫$(1,3)$▫-trees, that is, trees with all of their vertice shaving valency 1 or 3, thus extending the classical result of Frucht, Graver and Watkins (see [R. Frucht, J.E. Graver, M.E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc. 70 (1971) 211-218]) about arc-transitive strong expansions of ▫$K_2$▫ (also known as the generalized Petersen graphs). In this paper another step is taken further by considering the possible structure of cubic vertex-transitive expansions of general ▫$(1,3)$▫-multitrees (where vertices with double edges are also allowed); thus the restriction on every leaf to be expanded to a single cycle is dropped.
Found in: ključnih besedah
Summary of found: ...single cycle in ▫$X$▫. They determined all cubic arc-transitive strong expansions of simple ▫$(1,3)$▫-trees, that...
Keywords: graph, tree, cubic, vertex-transitive, arc-transitive, expansion
Published: 15.10.2013; Views: 2264; Downloads: 56
URL Full text (0,00 KB)

2.
A complete classification of cubic symmetric graphs of girth 6
Klavdija Kutnar, Dragan Marušič, 2009, original scientific article

Abstract: A complete classification of cubic symmetric graphs of girth 6 is given. It is shown that with the exception of the Heawood graph, the Moebius-Kantor graph, the Pappus graph, and the Desargues graph, a cubic symmetric graph ▫$X$▫ of girth 6 is a normal Cayley graph of a generalized dihedral group; in particular, (i) ▫$X$▫ is 2-regular if and only if it is isomorphic to a so-called ▫$I_k^n$▫-path, a graph of order either ▫$n^2/2$▫ or ▫$n^2/6$▫, which is characterized by the fact that its quotient relative to a certain semiregular automorphism is a path. (ii) ▫$X$▫ is 1-regular if and only if there exists an integer ▫$r$▫ with prime decomposition ▫$r=3^s p_1^{e_1} \dots p_t^{e_t} > 3$▫, where ▫$s \in \{0,1\}$▫, ▫$t \ge 1$▫, and ▫$p_i \equiv 1 \pmod{3}$▫, such that ▫$X$▫ is isomorphic either to a Cayley graph of a dihedral group ▫$D_{2r}$▫ of order ▫$2r$▫ or ▫$X$▫ is isomorphic to a certain ▫$\ZZ_r$▫-cover of one of the following graphs: the cube ▫$Q_3$▫, the Pappus graph or an ▫$I_k^n(t)$▫-path of order ▫$n^2/2$▫.
Found in: ključnih besedah
Summary of found: Zadetek v naslovu
Keywords: graph theory, cubic graphs, symmetric graphs, ▫$s$▫-regular graphs, girth, consistent cycle
Published: 15.10.2013; Views: 2046; Downloads: 59
URL Full text (0,00 KB)

3.
On 2-fold covers of graphs
Yan-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.
Found in: ključnih besedah
Summary of found: ...considered, with a particular emphasis given to cubic symmetric graphs. Supposing that ▫$G$▫ is transitive...
Keywords: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection
Published: 15.10.2013; Views: 1643; Downloads: 15
URL Full text (0,00 KB)

4.
Classification of cubic symmetric tetracirculants and pentacirculants
Boštjan Frelih, Klavdija Kutnar, 2013, original scientific article

Found in: ključnih besedah
Summary of found: Zadetek v naslovu
Keywords: cubic graph, symmetric, semiregular automorphism, tetracirculant, pentacirculant
Published: 15.10.2013; Views: 1654; Downloads: 58
URL Full text (0,00 KB)

5.
Cubic symmetric graphs via odd automorphisms, 60th Birthday Lecture Series, Department of Mathematics, University of Auckland, New Zealand, 10 September 2015
Klavdija Kutnar, 2015, invited lecture at foreign university

Found in: ključnih besedah
Summary of found: Zadetek v naslovu
Keywords: cubic graph, symmetric, automorphism, odd permutation
Published: 15.10.2015; Views: 1216; Downloads: 10
URL Full text (0,00 KB)

6.
An approach to geometric interpolation by Pythagorean-hodograph curves
Vito Vitrih, Marjetka Krajnc, Emil Žagar, Jernej Kozak, Gašper Jaklič, 2012, original scientific article

Abstract: The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree ▫$n$▫ is studied independently of the dimension ▫$d \ge 2$▫. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.
Found in: ključnih besedah
Summary of found: ...The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some...
Keywords: mathematics, parametric curve, PH curve, geometric interpolation, Lagrange interpolation, Hermite interpolation, cubic curves, homotopy
Published: 03.04.2017; Views: 758; Downloads: 43
URL Full text (0,00 KB)

7.
C [sup] 1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs
Miroslav Lávička, Zbiněk Šír, Vito Vitrih, Marjetka Krajnc, Bohumír Bastl, Michal Bizzarri, Emil Žagar, Kristýna Michálkova, 2014, original scientific article

Abstract: In this paper the ▫$C^1$▫ Hermite interpolation problem by spatial Pythagorean-hodograph cubic biarcs is presented and a general algorithm to construct such interpolants is described. Each PH cubic segment interpolates ▫$C^1$▫ data at one point and they are then joined together with a ▫$C^1$▫ continuity at some unknown common point sharing some unknown tangent vector. Biarcs are expressed in a closed form with three shape parameters. Two of them are selected based on asymptotic approximation order, while the remaining one can be computed by minimizing the length of the biarc or by minimizing the elastic bending energy. The final interpolating spline curve is globally ▫$C^1$▫ continuous, it can be constructed locally and it exists for arbitrary Hermite data configurations.
Found in: ključnih besedah
Summary of found: Zadetek v naslovu
Keywords: mathematics, parametric curve, PH curve, Pythagorean-hodograph, Hermite interpolation, biarc, cubic curve
Published: 03.04.2017; Views: 722; Downloads: 13
URL Full text (0,00 KB)

8.
Odd extensions of transitive groups via symmetric graphs - The cubic case
Klavdija Kutnar, Dragan Marušič, 2018, original scientific article

Abstract: When dealing with symmetry properties of mathematical objects, one of the fundamental questions is to determine their full automorphism group. In this paper this question is considered in the context of even/odd permutations dichotomy. More precisely: when is it that the existence of automorphisms acting as even permutations on the vertex set of a graph, called even automorphisms, forces the existence of automorphisms that act as odd permutations, called odd automorphisms. As a first step towards resolving the above question, complete information on the existence of odd automorphisms in cubic symmetric graphs is given.
Found in: ključnih besedah
Summary of found: Zadetek v naslovu
Keywords: automorphism group, arc-transitive, even permutation, odd permutation, cubic symmetric graph
Published: 19.11.2018; Views: 460; Downloads: 93
URL Full text (0,00 KB)

9.
Quasi-semiregular automorphisms of cubic and tetravalent arc-transitive graphs
István Kovács, Yan-Quan Feng, Ademir Hujdurović, Klavdija Kutnar, Dragan Marušič, 2018, unpublished conference contribution

Found in: ključnih besedah
Summary of found: Zadetek v naslovu
Keywords: quasi-semiregular automorphism, cubic graph, tetravalent graph, arc-transitive graph
Published: 06.12.2018; Views: 463; Downloads: 67
URL Full text (0,00 KB)

10.
Quasi-semiregular automorphisms of cubic and tetravalent arc-transitive graphs
Yan-Quan Feng, Ademir Hujdurović, István Kovács, Klavdija Kutnar, Dragan Marušič, 2019, original scientific article

Found in: ključnih besedah
Summary of found: Zadetek v naslovu
Keywords: cubic graph, tetravalent graph, arc-transitive, quasi-semiregular automorphism
Published: 14.03.2019; Views: 370; Downloads: 110
URL Full text (0,00 KB)

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