Distance-regular Cayley graphs on dihedral groups

Štefko Miklavič (Author), Primož Potočnik (Author)

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;diedrska grupa;diferenčna množica;mathematics;grah theory;distance-regular graph;distance-transitive graph;Cayley graph;dihedral group;dihedrant;difference set;

Data

Language:	English
Year of publishing:	2005
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	519.17
COBISS:	13784665
ISSN:	1318-4865
Views:	61
Downloads:	15
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Razdaljno regularni Cayleyjevi grafi na diedrskih grupah
Secondary 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 dierence 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.
Secondary keywords:	matematika;teorija grafov;Cayleyjev graf;razdaljno-regularen graf;razdaljno-trazitiven graf;Cayleyjev graf;diedrska grupa;diferenčna množica;
Type (COBISS):	Not categorized
Pages:	str. 1-27
Volume:	Vol. 43
Issue:	št. 989
Chronology:	2005
ID:	56423