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31.
Strongly regular tri-Cayley graphs
Klavdija Kutnar, Dragan Marušič, Štefko Miklavič, Primož Šparl, 2009, original scientific article

Abstract: A graph is called tri-Cayley if it admits a semiregular subgroup of automorphisms having three orbits of equal length. In this paper, the structure of strongly regular tri-Cayley graphs is investigated. A structural description of strongly regular tri-Cayley graphs of cyclic groups is given.
Found in: ključnih besedah
Summary of found: ...A graph is called tri-Cayley if it admits a...
Keywords: strongly regular graph, tri-Cayley graph
Published: 15.10.2013; Views: 1226; Downloads: 47
URL Full text (0,00 KB)

32.
Distance-regular Cayley graphs on dihedral groups
Štefko 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.
Found in: ključnih besedah
Summary of found: ...article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious...
Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set
Published: 15.10.2013; Views: 1231; Downloads: 53
URL Full text (0,00 KB)

33.
A note on domination and independence-domination numbers of graphs
Martin Milanič, 2013, published scientific conference contribution

Abstract: Vizing's conjecture is true for graphs ▫$G$▫ satisfying ▫$\gamma^i(G) = \gamma(G)$▫, where ▫$\gamma(G)$▫ is the domination number of a graph ▫$G$▫ and ▫$\gamma^i(G)$▫ is the independence-domination number of ▫$G$▫, that is, the maximum, over all independent sets ▫$I$▫ in ▫$G$▫, of the minimum number of vertices needed to dominate ▫$I$▫. The equality ▫$\gamma^i(G) = \gamma(G)$▫ is known to hold for all chordal graphs and for chordless cycles of length ▫$0 \pmod{3}$▫. We prove some results related to graphs for which the above equality holds. More specifically, we show that the problems of determining whether ▫$\gamma^i(G) = \gamma(G) = 2$▫ and of verifying whether ▫$\gamma^i(G) \ge 2$▫ are NP-complete, even if ▫$G$▫ is weakly chordal. We also initiate the study of the equality ▫$\gamma^i = \gamma$▫ in the context of hereditary graph classes and exhibit two infinite families of graphs for which ▫$\gamma^i < \gamma$▫.
Found in: ključnih besedah
Summary of found: ...Vizing's conjecture is true for graphs ▫$G$▫ satisfying ▫$\gamma^i(G) = \gamma(G)$▫, where ▫$\gamma(G)$▫...
Keywords: Vizing's conjecture, domination number, independence-domination number, weakly chordal graph, NP-completeness, hereditary graph class, IDD-perfect graph
Published: 15.10.2013; Views: 1226; Downloads: 62
URL Full text (0,00 KB)

34.
Hamiltonicity of cubic Cayley graphs
Dragan Marušič, Henry Glover, Klavdija Kutnar, Aleksander Malnič, 2012, published scientific conference contribution abstract (invited lecture)

Found in: ključnih besedah
Summary of found: ...Cayley graph, Hamilton path, Hamilton cycle, arc-transitive graph, Cayley...
Keywords: Cayley graph, Hamilton path, Hamilton cycle, arc-transitive graph, Cayley map
Published: 15.10.2013; Views: 1360; Downloads: 25
URL Full text (0,00 KB)

35.
Construction of Hamilton cycles in (2,s,3)-Cayley graphs
Klavdija Kutnar, 2010, published scientific conference contribution abstract

Found in: ključnih besedah
Summary of found: ...Hamilton cycle, Cayley graph, ...
Keywords: Hamilton cycle, Cayley graph
Published: 15.10.2013; Views: 1299; Downloads: 8
URL Full text (0,00 KB)

36.
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: ... graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular...
Keywords: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection
Published: 15.10.2013; Views: 1226; Downloads: 13
URL Full text (0,00 KB)

37.
Asymptotic automorphism groups of Cayley digraphs and graphs of abelian groups of prime-power order
Edward Dobson, 2010, original scientific article

Abstract: We show that almost every Cayley graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of ▫$G$▫ (that is, ▫$G_L \triangleleft {\rm Aut}(\Gamma))$▫.
Found in: ključnih besedah
Summary of found: ...We show that almost every Cayley graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of...
Keywords: mathematics, graph theory, Cayley graph, abelian group, automorphism group, asymptotic, ▫$p$▫-group
Published: 15.10.2013; Views: 2437; Downloads: 54
URL Full text (0,00 KB)

38.
On generalized Cayley graphs
Klavdija Kutnar, 2013, published scientific conference contribution abstract

Found in: ključnih besedah
Summary of found: ...vertex-transitive, bicirculant, generalized Cayley graph, ...
Keywords: vertex-transitive, bicirculant, generalized Cayley graph
Published: 15.10.2013; Views: 1869; Downloads: 11
URL Full text (0,00 KB)

39.
Hamilton paths and cycles in vertex-transitive graphs of order 6p
Klavdija Kutnar, Primož Šparl, 2009, original scientific article

Abstract: It is shown that every connected vertex-transitive graph of order ▫$6p$▫, where ▫$p$▫ is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order ▫$6p$▫ which is not genuinely imprimitive contains a Hamilton cycle.
Found in: ključnih besedah
Summary of found: ...It is shown that every connected vertex-transitive graph of order ▫$6p$▫, where ▫$p$▫ is a...
Keywords: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group
Published: 15.10.2013; Views: 1626; Downloads: 14
URL Full text (0,00 KB)

40.
Quasi m-Cayley circulants
Ademir Hujdurović, 2013, published scientific conference contribution

Abstract: A graph ▫$\Gamma$▫ is called a quasi ▫$m$▫-Cayley graph on a group ▫$G$▫ if there exists a vertex ▫$\infty \in V(\Gamma)$▫ and a subgroup ▫$G$▫ of the vertex stabilizer ▫$\text{Aut}(\Gamma)_\infty$▫ of the vertex ▫$\infty$▫ in the full automorphism group ▫$\text{Aut}(\Gamma)$▫ of ▫$\Gamma$▫, such that ▫$G$▫ acts semiregularly on ▫$V(\Gamma) \setminus \{\infty\}$▫ with ▫$m$▫ orbits. If the vertex ▫$\infty$▫ is adjacent to only one orbit of ▫$G$▫ on ▫$V(\Gamma) \setminus \{\infty\}$▫, then ▫$\Gamma$▫ is called a strongly quasi ▫$m$▫-Cayley graph on ▫$G$▫ .In this paper complete classifications of quasi 2-Cayley, quasi 3-Cayley and strongly quasi 4-Cayley connected circulants are given.
Found in: ključnih besedah
Summary of found: ...A graph ▫$\Gamma$▫ is called a quasi ▫$m$▫-Cayley graph...
Keywords: arc-transitive, circulant, quasi m-Cayley graph
Published: 15.10.2013; Views: 1642; Downloads: 54
URL Full text (0,00 KB)

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