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2. Testing whether the lifted group splitsRok Požar, 2016, original scientific article Abstract: Let a group of automorphisms lift along a regular covering projection of connected graphs given combinatorially by means of voltages. The data that determine the lifted group and its action are then conveniently encoded in terms of voltages as well. Along these lines, an algorithm for testing whether the lifted group is a split extension of the group of covering transformations has recently been proposed in the case when the group of covering transformations is solvable. It consists of decomposing the covering into a series of coverings with elementary abelian groups of covering transformations, and inductively solving the problem at every elementary abelian step. Although the explicit construction of the lifted group is not needed, it still involves time and space consuming constructions of certain subgroups in the lifted group at every step except at the final one. In this paper, an improved version that completely avoids such constructions is presented. From voltage distribution we first compute the weak action and the factor set that determine the lifted group, and we then carry out the test by extracting the necessary information only from the corresponding weak actions and factor sets at every step. An experimental comparison is made against the previous version. Keywords: algorithm, graph, group extension, lifting automorphisms, regular covering projection, voltages Published in RUP: 03.01.2022; Views: 840; Downloads: 17 Full text (317,95 KB) |
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9. Algebraični aspekti teorije grafov : doktorska disertacijaAdemir Hujdurović, 2013, doctoral dissertation Keywords: circulant, bicirculant, semiregular automorphism, vertex-transitive graph, half-arc-transitive graph, snark, Cayley graph, quasi m-Cayley graph, generalized Cayley graph, I-regular action, regular cover of a graph, automorphism group Published in RUP: 10.07.2015; Views: 3536; Downloads: 42 Link to full text |
10. Distance-regular Cayley graphs on dihedral groupsŠtefko Miklavič, Primož Potočnik, 2005, 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. Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set Published in RUP: 10.07.2015; Views: 2631; Downloads: 90 Link to full text |