Distance-regular Cayley graphs on dihedral groups

Miklavič, Štefko; Potočnik, Primož

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Title:	Distance-regular Cayley graphs on dihedral groups
Authors:	ID Miklavič, Štefko (Author) ID Potočnik, Primož (Author)
Files:	http://dx.doi.org/10.1016/j.jctb.2006.03.003
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
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
Year of publishing:	2007
Number of pages:	str. 14-33
Numbering:	Vol. 97, no. 1
PID:	20.500.12556/RUP-2594
ISSN:	0095-8956
UDC:	519.17
COBISS.SI-ID:	1909207
Publication date in RUP:	15.10.2013
Views:	4021
Downloads:	103
Metadata:
:	MIKLAVIČ, Štefko and POTOČNIK, Primož, 2007, Distance-regular Cayley graphs on dihedral groups. [online]. 2007. Vol. 97, no. 1, p. 14–33. [Accessed 28 March 2025]. Retrieved from: http://dx.doi.org/10.1016/j.jctb.2006.03.003 Copy citation

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Secondary language

Language:	Slovenian
Abstract:	Glavni rezultat tega članka je klasifikacija razdaljno-regularnih Cayleyjevih grafov na diedrskih grupah. Naslednje štiri družine takšnih razdaljno-regularnih grafov bomo imenovali trivialne: polni grafi, polni dvodelni grafi, polni dvodelni grafi brez 1-faktorja in cikli. V članku dokažemo, da je vsak netrivialen Cayleyjev razdaljno-regularen graf na diedrski grupi dvodelen, neantipoden, premera 3, ter da je porojen iz ciklične diferenčne množice ali iz diedrske diferenčne množice, ki zadošča nekaterim dodatnim pogojem (če kakšna taka sploh obstaja). Poiščemo tudi vse Cayleyeve razdaljno-tranzitivne grafe na diedrskih grupah. Izkaže se, da je Cayleyjev graf na diedrski grupi razdaljno-tranzitiven natanko takrat ko je trivialen, ali pa izomorfen bodisi incidenčnemu bodisi neincidenčnemu grafu projektivnega prostora $\mathrm{PG}_{d-1} (d,q)$ , $d \ge 2$ , ali enolično določenega komplementarnega para simetričnih načrtov na enajstih točkah.
Keywords:	matematika, teorija grafov, Cayleyjev graf, razdaljno-regularen graf, razdaljno-trazitiven graf, Cayleyjev graf, diedrska grupa, diferenčna množica

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