11. Asymptotic automorphism groups of Cayley digraphs and graphs of abelian groups of prime-power orderEdward 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 Keywords: mathematics, graph theory, Cayley graph, abelian group, automorphism group, asymptotic, ▫$p$▫-group Published: 15.10.2013; Views: 2521; Downloads: 58 Full text (0,00 KB) |
12. Hamilton paths and cycles in vertex-transitive graphs of order 6pKlavdija 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: ...graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group... Keywords: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Published: 15.10.2013; Views: 1676; Downloads: 14 Full text (0,00 KB) |
13. On cubic non-Cayley vertex-transitive graphsKlavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, original scientific article Found in: ključnih besedah Summary of found: ...vertex-transitive graph, non-Cayley graph, automorphism group, ... Keywords: vertex-transitive graph, non-Cayley graph, automorphism group Published: 15.10.2013; Views: 1270; Downloads: 63 Full text (0,00 KB) |
14. Hamilton cycles in (2, odd, 3)-Cayley graphsHenry Glover, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2012, original scientific article Abstract: In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite ▫$(2, s, 3)$▫-generated group ▫$G = \langle a, x| a^2 = x^s = (ax)^3 = 1, \dots \rangle$▫ has a Hamilton path when ▫$|G|$▫ is congruent to 0 modulo 4, and has a Hamilton cycle when ▫$|G|$▫ is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from ▫$|G|$▫, also ▫$s$▫ is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever ▫$|G|$▫ is congruent to 0 modulo 4 and s is odd. This leaves ▫$|G|$▫ congruent to 0 modulo 4 with s congruent to 2 modulo 4 as the only remaining open case. In this last case, however, the "tree of faces" approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in cubic Cayley graphs arising from finite ▫$(2, s, 3)$▫-generated groups. Found in: ključnih besedah Summary of found: ...graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group... Keywords: Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group Published: 15.10.2013; Views: 1249; Downloads: 60 Full text (0,00 KB) |
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17. On vertex-stabilizers of bipartite dual polar graphsŠtefko Miklavič, 2010, original scientific article Abstract: Let ▫$X,Y$▫ denote vertices of a bipartite dual polar graph, and let ▫$G_X$▫ and ▫$G_Y$▫ denote the stabilizers of ▫$X$▫ and ▫$Y$▫ in the full automorphism group of this graph. In this paper, a description of the orbits of ▫$G_X \cap G_Y$▫ in the cases when the distance between ▫$X$▫ and ▫$Y$▫ is 1 or 2, is given. Found in: ključnih besedah Summary of found: ...of ▫$X$▫ and ▫$Y$▫ in the full automorphism group of this graph. In this paper,... Keywords: dual polar graphs, automorphism group, quadratic form, isotropic subspace Published: 15.10.2013; Views: 1375; Downloads: 57 Full text (0,00 KB) |
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19. 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: 1394; Downloads: 50 Full text (0,00 KB) |
20. Distance-transitive graphs admit semiregular automorphismsKlavdija Kutnar, Primož Šparl, 2010, original scientific article Abstract: 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 besedah Summary of found: ...between ▫$u'$▫ and ▫$v'$▫ there exists an automorphism of the graph mapping ▫$u$▫ to ▫$u'$▫... Keywords: distance-transitive graph, vertex-transitive graph, semiregular automorphism, permutation group Published: 15.10.2013; Views: 1696; Downloads: 57 Full text (0,00 KB) |