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: ...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: 1997; Downloads: 52 Full text (0,00 KB) |
2. Classification of cubic symmetric tricirculantsIstván Kovács, Klavdija Kutnar, Dragan Marušič, Stephen Wilson, 2012, original scientific article Found in: ključnih besedah Summary of found: ...symmetric graph, semiregular, tricirculant, ... Keywords: symmetric graph, semiregular, tricirculant Published: 15.10.2013; Views: 1623; Downloads: 25 Full text (0,00 KB) |
3. The Terwilliger algebra of a distance-regular graph of negative typeŠtefko Miklavič, 2009, original scientific article Abstract: Let ▫$\Gamma$▫ denote a distance-regular graph with diameter ▫$D \ge 3$▫. Assume ▫$\Gamma$▫ has classical parameters ▫$(D,b,\alpha,\beta)▫$ with ▫$b < -1$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫ and let ▫$A \in {\mathrm{Mat}}_X(\mathbb{C})$▫ denote the adjacency matrix of ▫$\Gamma$▫. Fix ▫$x \in X$▫ and let $A^\ast \in {\mathrm{Mat}}_X(\mathbb{C})$ denote the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ${\mathrm{Mat}}_X(\mathbb{C})$ generated by ▫$A,A^\ast$▫. We call ▫$T$▫ the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. We show that up to isomorphism there exist exactly two irreducible ▫$T$▫-modules with endpoint 1; their dimensions are ▫$D$▫ and ▫$2D-2$▫. For these ▫$T$▫-modules we display a basis consisting of eigenvectors for ▫$A^\ast$▫, and for each basis we give the action of ▫$A$▫. Found in: ključnih besedah Keywords: distance-regular graph, negative type, Terwilliger algebra Published: 15.10.2013; Views: 1676; Downloads: 77 Full text (0,00 KB) |
4. Hamilton paths in vertex-transitive graphs of order 10pKlavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, original scientific article Abstract: 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 besedah Summary of found: ...It is shown that every connected vertex-transitive graph of order ▫$10p$▫, ▫$p \ne 7$▫ a... Keywords: graph, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Published: 15.10.2013; Views: 1772; Downloads: 12 Full text (0,00 KB) |
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6. On prime-valent symmetric bicirculants and Cayley snarksAdemir Hujdurović, Klavdija Kutnar, Dragan Marušič, 2013, published scientific conference contribution Found in: ključnih besedah Summary of found: ... graph, Cayley graph, arc-transitive, snark, semiregular automorphism, bicirculant,... Keywords: graph, Cayley graph, arc-transitive, snark, semiregular automorphism, bicirculant Published: 15.10.2013; Views: 1629; Downloads: 81 Full text (0,00 KB) |
7. Hamiltonicity of vertex-transitive graphs of order 4pKlavdija Kutnar, Dragan Marušič, 2008, original scientific article Abstract: 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 besedah Summary of found: ...It is shown that every connected vertex-transitive graph of order ▫$4p$▫, where ▫$p$▫ is a... Keywords: graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group Published: 15.10.2013; Views: 1649; Downloads: 18 Full text (0,00 KB) |
8. Isomorphism checking of I-graphsBoris Horvat, Tomaž Pisanski, Arjana Žitnik, 2012, original scientific article Abstract: We consider the class of ▫$I$▫-graphs, which is a generalization of the class of the generalized Petersen graphs. We show that two ▫$I$▫-graphs ▫$I(n, j, k)$▫ and ▫$I(n, j_1, k_1)$▫ are isomorphic if and only if there exists an integer ▫$a$▫ relatively prime to $n$ such that either ▫$\{j_1, k_1\} = \{aj \mod n, \; ak \mod n \}$▫ or ▫$\{j_1, k_1\} = \{aj \mod n, \; -ak \mod n\}$▫. This result has an application in the enumeration of non-isomorphic ▫$I$▫-graphs and unit-distance representations of generalized Petersen graphs. Found in: ključnih besedah Summary of found: ...mathematics, graph theory, isomorphism, I-graph, generalized Petersen graph... Keywords: mathematics, graph theory, isomorphism, I-graph, generalized Petersen graph Published: 15.10.2013; Views: 1599; Downloads: 76 Full text (0,00 KB) |
9. On hereditary efficiently dominatable graphsMartin Milanič, 2011, unpublished conference contribution Found in: ključnih besedah Summary of found: ...perfect code, efficient domination, efficiently dominatable graph, polynomial time algorithm, ... Keywords: perfect code, efficient domination, efficiently dominatable graph, polynomial time algorithm Published: 15.10.2013; Views: 1439; Downloads: 42 Full text (0,00 KB) |
10. Distance-balanced graphs: Symmetry conditionsKlavdija Kutnar, Aleksander Malnič, Dragan Marušič, Štefko Miklavič, 2006, original scientific article Abstract: 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 besedah Summary of found: ...A graph ▫$X$▫ is said to be distance-balanced if... Keywords: graph theory, graph, distance-balanced graphs, vertex-transitive, semysimmetric, generalized Petersen graph Published: 15.10.2013; Views: 1928; Downloads: 55 Full text (0,00 KB) |