1. Vertex-transitive expansions of (1, 3)-treesMarko 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... ...A nonidentity automorphism of a graph is said to be semiregular if all... Keywords: graph, tree, cubic, vertex-transitive, arc-transitive, expansion Published: 15.10.2013; Views: 1995; Downloads: 52 Full text (0,00 KB) |
2. A complete classification of cubic symmetric graphs of girth 6Klavdija 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: ...A complete classification of cubic symmetric graphs of girth 6 is given. It is... Keywords: graph theory, cubic graphs, symmetric graphs, ▫$s$▫-regular graphs, girth, consistent cycle Published: 15.10.2013; Views: 1850; Downloads: 52 Full text (0,00 KB) |
3. 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. 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: 1326; Downloads: 14 Full text (0,00 KB) |
4. Classification of cubic symmetric tetracirculants and pentacirculantsBoštjan Frelih, Klavdija Kutnar, 2013, original scientific article Found in: ključnih besedah Summary of found: ...cubic graph, symmetric, semiregular automorphism, tetracirculant, pentacirculant, ... Keywords: cubic graph, symmetric, semiregular automorphism, tetracirculant, pentacirculant Published: 15.10.2013; Views: 1438; Downloads: 53 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 2015Klavdija Kutnar, 2015, invited lecture at foreign university Found in: ključnih besedah Summary of found: ...cubic graph, symmetric, automorphism, odd permutation, ... Keywords: cubic graph, symmetric, automorphism, odd permutation Published: 15.10.2015; Views: 1087; Downloads: 10 Full text (0,00 KB) |
6. Odd extensions of transitive groups via symmetric graphs - The cubic caseKlavdija 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: ...permutations on the vertex set of a graph, called even automorphisms, forces the existence of... Keywords: automorphism group, arc-transitive, even permutation, odd permutation, cubic symmetric graph Published: 19.11.2018; Views: 323; Downloads: 83 Full text (0,00 KB) |
7. Quasi-semiregular automorphisms of cubic and tetravalent arc-transitive graphsIstván Kovács, Yan-Quan Feng, Ademir Hujdurović, Klavdija Kutnar, Dragan Marušič, 2018, unpublished conference contribution Found in: ključnih besedah Summary of found: ...quasi-semiregular automorphism, cubic graph, tetravalent graph, arc-transitive graph... Keywords: quasi-semiregular automorphism, cubic graph, tetravalent graph, arc-transitive graph Published: 06.12.2018; Views: 327; Downloads: 57 Full text (0,00 KB) |
8. Quasi-semiregular automorphisms of cubic and tetravalent arc-transitive graphsYan-Quan Feng, Ademir Hujdurović, István Kovács, Klavdija Kutnar, Dragan Marušič, 2019, original scientific article Found in: ključnih besedah Summary of found: ...cubic graph, tetravalent graph, arc-transitive, quasi-semiregular automorphism... Keywords: cubic graph, tetravalent graph, arc-transitive, quasi-semiregular automorphism Published: 14.03.2019; Views: 255; Downloads: 96 Full text (0,00 KB) |
9. Detecting strong cliquesAdemir Hujdurović, Martin Milanič, Bernard Ries, 2019, original scientific article Found in: ključnih besedah Summary of found: ...strong clique, weakly chordal graph, line graph, cubic graph... Keywords: strong clique, weakly chordal graph, line graph, cubic graph Published: 30.06.2019; Views: 436; Downloads: 87 Full text (0,00 KB) |