On strongly regular bicirculants

  • Authors:

  • Aleksander Malnič;Dragan Marušič;Primo Šparl

  • Affiliations:

  • IMFM, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia;University of Primorska, Cankarjeva 5, 6000 Koper, Slovenia and IMFM, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia;IMFM, University of Ljubljana, Jadranska 19, 1111 Ljubljana, Slovenia

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

An n-bicirculant is 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=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.