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: 3250; Downloads: 76 Full text (0,00 KB) |
2. Classification of cubic symmetric tricirculantsKlavdija Kutnar, István Kovács, Dragan Marušič, Steve 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: 2751; Downloads: 76 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: 2736; Downloads: 105 Full text (0,00 KB) |
4. Hamilton paths in vertex-transitive graphs of order 10pDragan Marušič, Klavdija Kutnar, 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: 2962; Downloads: 36 Full text (0,00 KB) |
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6. On prime-valent symmetric bicirculants and Cayley snarksKlavdija Kutnar, Ademir Hujdurović, 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: 2727; Downloads: 148 Full text (0,00 KB) |
7. A complexitystudy of distance variants of covering and domination problems in H-free graphsMirza Krbezlija, 2021, master's thesis Found in: ključnih besedah Summary of found: ...distance-k vertex cover, distance k-edge cover, H-free graph, polynomial-time algorithm, NP-complete problem, dichotomy theorem... Keywords: distance-k dominating set, distance-k edge dominating set, distance-k vertex cover, distance k-edge cover, H-free graph, polynomial-time algorithm, NP-complete problem, dichotomy theorem Published: 18.10.2021; Views: 464; Downloads: 15 Full text (0,00 KB) |
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9. 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: 2876; Downloads: 34 Full text (0,00 KB) |
10. On cubic arc-transitive k-multicirculants with soluble groupsBoštjan Frelih, István Kovács, Klavdija Kutnar, 2021, original scientific article Found in: ključnih besedah Summary of found: ...cubic graph, k-multicirculant, arc-transitive, soluble group... Keywords: cubic graph, k-multicirculant, arc-transitive, soluble group Published: 26.10.2021; Views: 323; Downloads: 17 Full text (0,00 KB) |