On the existence and the enumeration of bipartite regular representations of Cayley graphs over abelian groups

In this paper, we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let (Formula presented.) be an abelian group and let (Formula presented.) be the automorphism of (Formula presented.) defined by (Formula presented.), for every (Formula presented.). A Cayley graph (Formula presented.) is said to have an automorphism group as small as possible if (Formula presented.). In this paper, we show that, except for two infinite families, almost all bipartite Cayley graphs on abelian groups have automorphism group as small as possible. We also investigate the analogous question for bipartite Cayley digraphs. These results are used for the asymptotic enumeration of bipartite Cayley digraphs and graphs over abelian groups.