On strongly regular bicirculants

Aleksander Malnič (Author), Dragan Marušič (Author), Primož Šparl (Author)

Abstract

An ▫$n$▫-bicirculantis a graph having an automorphism with two orbits of length ▫$n$▫ and no other orbits. This article deals with strongly regular bicirculants. It is known that for a nontrivial strongly regular ▫$n$▫-bicirculant, ▫$n$▫ odd, there exists a positive integer m such that ▫$n=2m^2+2m+1▫$. Only three nontrivial examples have been known previously, namely, for ▫$m=1,2$▫ and 4. Case ▫$m=1$▫ gives rise to the Petersen graph and its complement, while the graphs arising from cases ▫$m=2$▫ and ▫$m=4$▫ are associated with certain Steiner systems. Similarly, if ▫$n$▫ is even, then ▫$n=2m^2$▫ for some ▫$m \ge 2$▫. Apart from a pair of complementary strongly regular 8-bicirculants, no other example seems to be known. A necessary condition for the existence of a strongly regular vertex-transitive ▫$p$▫-bicirculant, ▫$p$▫ a prime, is obtained here. In addition, three new strongly regular bicirculants having 50, 82 and 122 vertices corresponding, respectively, to ▫$m=3,4$▫ and 5 above, are presented. These graphs are not associated with any Steiner system, and together with their complements form the first known pairs of complementary strongly regular bicirculants which are vertex-transitive but not edge-transitive.

Keywords

matematika;teorija grafov;graf;cirkulant;bicirkulant;grupa avtomorfizmov;mathematics;graph theory;graph;circulant;bicirculant;automorphism group;

Data

Language:	English
Year of publishing:	2007
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	519.17:512.54
COBISS:	14287705
ISSN:	0195-6698
Views:	50
Downloads:	29
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary keywords:	matematika;teorija grafov;graf;cirkulant;bicirkulant;grupa avtomorfizmov;
Type (COBISS):	Not categorized
Pages:	str. 891-900
Volume:	Vol. 28
Issue:	iss. 3
Chronology:	2007
ID:	1473065