In this paper we are interested in the asymptotic enumeration of bipartite Cayley digraphs and Cayley graphs over abelian groups. Let $A$ be an abelian group and let $\iota$ be the automorphism of $A$ defined by $a^\iota=a^{-1}$, for every $a\in A$. A Cayley graph $\Cay(A, S)$ is said to have an automorphism group as small as possible if $\Aut(\Cay(A,S)) = \langle A,\iota\rangle$. 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.

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关于阿贝尔群上凯莱图的二部正则表示的存在和枚举

在本文中，我们对二部 Cayley 有向图和 Cayley 图在阿贝尔群上的渐近枚举感兴趣。令 $A$ 是一个阿贝尔群，令 $\iota$ 是 $a^\iota=a^{-1}$ 定义的 $A$ 的自同构，对于每个 $a\in A$。如果 $\Aut(\Cay(A,S)) = \langle A,\iota\rangle$，则称凯莱图 $\Cay(A, S)$ 具有尽可能小的自同构群。在本文中，我们表明，除了两个无限族外，几乎所有阿贝尔群上的二部凯莱图都有尽可能小的自同构群。我们还研究了二部凯莱有向图的类似问题。