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 1 - 10 / 401234 1.Vertex-transitive expansions of (1, 3)-treesMarko Lovrečič Saražin, Dragan Marušič, 2010, published scientific conference contributionAbstract: 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 besedahSummary of found: Zadetek v naslovuKeywords: graph, tree, cubic, vertex-transitive, arc-transitive, expansionPublished: 15.10.2013; Views: 2211; Downloads: 55 Full text (0,00 KB) 2.Hamilton paths in vertex-transitive graphs of order 10pKlavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, original scientific articleAbstract: It is shown that every connected vertex-transitive graph of order ▫$10p$▫, ▫$p \ne 7$▫ a prime, which is not isomorphic to a quasiprimitive graph arising from the action of PSL▫$(2,k)$▫ on cosets of ▫$\mathbb{Z}_k \times \mathbb{Z}_{(k-1)/10}$▫, contains a Hamilton path.Found in: ključnih besedahSummary of found: Zadetek v naslovuKeywords: graph, vertex-transitive, Hamilton cycle, Hamilton path, automorphism groupPublished: 15.10.2013; Views: 1967; Downloads: 12 Full text (0,00 KB) 3.Hamiltonicity of vertex-transitive graphs of order 4pKlavdija Kutnar, Dragan Marušič, 2008, original scientific articleAbstract: It is shown that every connected vertex-transitive graph of order ▫$4p$▫, where ▫$p$▫ is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path.Found in: ključnih besedahSummary of found: Zadetek v naslovuKeywords: graph theory, vertex-transitive graphs, Hamilton cycle, automorphism groupPublished: 15.10.2013; Views: 1856; Downloads: 18 Full text (0,00 KB) 4.Distance-balanced graphs: Symmetry conditionsKlavdija Kutnar, Aleksander Malnič, Dragan Marušič, Štefko Miklavič, 2006, original scientific articleAbstract: A graph ▫$X$▫ is said to be distance-balanced if for any edge ▫$uv$▫ of ▫$X$▫, the number of vertices closer to ▫$u$▫ than to ▫$v$▫ is equal to the number of vertices closer to ▫$v$▫ than to ▫$u$▫. A graph ▫$X$▫ is said to be strongly distance-balanced if for any edge ▫$uv$▫ of ▫$X$▫ and any integer ▫$k$▫, the number of vertices at distance ▫$k$▫ from ▫$u$▫ and at distance ▫$k+1$▫ from ▫$v$▫ is equal to the number of vertices at distance ▫$k+1$▫ from ▫$u$▫ and at distance ▫$k$▫ from ▫$v$▫. Exploring the connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is the main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus also distance-balanced is an easy observation. With only a slight relaxation of the transitivity condition, the situation changes drastically: there are infinite families of semisymmetric graphs (that is, graphs which are edge-transitive, but not vertex-transitive) which are distance-balanced, but there are also infinite families of semisymmetric graphs which are not distance-balanced. Results on the distance-balanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distance-balanced graphs is given for the following infinite families of generalized Petersen graphs: GP▫$(n,2)$▫, GP▫$(5k+1,k)$▫, GP▫$(3k 3,k)$▫, and GP▫$(2k+2,k)$▫.Found in: ključnih besedahSummary of found: ...main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus... Keywords: graph theory, graph, distance-balanced graphs, vertex-transitive, semysimmetric, generalized Petersen graphPublished: 15.10.2013; Views: 2148; Downloads: 59 Full text (0,00 KB) 5.On generalized Cayley graphsKlavdija Kutnar, 2013, published scientific conference contribution abstractFound in: ključnih besedahSummary of found: ...vertex-transitive, bicirculant, generalized Cayley graph, ... Keywords: vertex-transitive, bicirculant, generalized Cayley graphPublished: 15.10.2013; Views: 2102; Downloads: 11 Full text (0,00 KB) 6.Hamilton paths and cycles in vertex-transitive graphs of order 6pKlavdija Kutnar, Primož Šparl, 2009, original scientific articleAbstract: 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 besedahSummary of found: Zadetek v naslovuKeywords: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism groupPublished: 15.10.2013; Views: 1905; Downloads: 15 Full text (0,00 KB) 7.Classification of edge-transitive rose window graphsIstván Kovács, Klavdija Kutnar, Dragan Marušič, 2010, original scientific articleAbstract: Given natural numbers ▫$n \ge 3$▫ and ▫$1 \le a$▫, ▫$r \le n-1$▫, the rose window graph ▫$R_n(a,r)$▫ is a quartic graph with vertex set ▫$\{x_i \vert i \in {\mathbb Z}_n\} \cup \{y_i \vert i \in {\mathbb Z}_n\}$▫ and edge set ▫$\{\{x_i, x_{i+1}\} \vert i \in {\mathbb Z}_n\} \cup \{\{y_i, y_{i+r}\} \vert i \in {\mathbb Z}_n\} \cup \{\{x_i, y_i\} \vert i \in {\mathbb Z}_n\} \cup \{\{x_{i+a}, y_i\} \vert i \in {\mathbb Z}_n\}$▫. In this article a complete classification of edge-transitive rose window graphs is given, thus solving one of three open problems about these graphs posed by Steve Wilson in 2001.Found in: ključnih besedahSummary of found: Zadetek v naslovuKeywords: group, graph, rose window, vertex-transitive, edge-transitive, arc-transitivePublished: 15.10.2013; Views: 1482; Downloads: 58 Full text (0,00 KB) 8.On cubic non-Cayley vertex-transitive graphsKlavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, original scientific articleFound in: ključnih besedahSummary of found: Zadetek v naslovuKeywords: vertex-transitive graph, non-Cayley graph, automorphism groupPublished: 15.10.2013; Views: 1487; Downloads: 73 Full text (0,00 KB) 9.On cubic non-Cayley vertex-transitive graphsKlavdija Kutnar, 2010, invited lecture at foreign universityFound in: ključnih besedahSummary of found: Zadetek v naslovuKeywords: non-Cayley, vertex-transitive, automorphism grupPublished: 15.10.2013; Views: 1670; Downloads: 52 Full text (0,00 KB) 10.Distance-transitive graphs admit semiregular automorphismsKlavdija Kutnar, Primož Šparl, 2010, original scientific articleAbstract: A distance-transitive graph is a graph in which for every two ordered pairs ofvertices ▫$(u,v)$▫ and ▫$(u',v')$▫ such that the distance between ▫$u$▫ and ▫$v$▫ is equal to the distance between ▫$u'$▫ and ▫$v'$▫ there exists an automorphism of the graph mapping ▫$u$▫ to ▫$u'$▫ and ▫$v$▫ to ▫$v'$▫. A semiregular element of a permutation group is anon-identity element having all cycles of equal length in its cycle decomposition. It is shown that every distance-transitive graph admits a semiregular automorphism.Found in: ključnih besedahSummary of found: Zadetek v naslovuKeywords: distance-transitive graph, vertex-transitive graph, semiregular automorphism, permutation groupPublished: 15.10.2013; Views: 1879; Downloads: 64 Full text (0,00 KB)
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