| Title: | Distance-regular Cayley graphs on dihedral groups |
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| Authors: | ID Miklavič, Štefko (Author)
ID Potočnik, Primož (Author) |
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| Files: |  http://dx.doi.org/10.1016/j.jctb.2006.03.003
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| Language: | English |
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| Work type: | Not categorized |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: | IAM - Andrej Marušič Institute
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| 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. |
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| Keywords: | mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set |
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| Year of publishing: | 2007 |
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| Number of pages: | str. 14-33 |
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| Numbering: | Vol. 97, no. 1 |
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| PID: | 20.500.12556/RUP-2594  |
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| ISSN: | 0095-8956 |
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| UDC: | 519.17 |
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| COBISS.SI-ID: | 1909207 |
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| Publication date in RUP: | 15.10.2013 |
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| Views: | 5508 |
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| Downloads: | 106 |
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