On strongly regular bicirculants

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 = 2 m 2 + 2 m + 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 = 2 m 2 for some m ≥ 2 . Apart from a pair of complementary strongly regular 8-bicirculants, no other example seems to be known. ssary 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.