A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. This generalisation of a Cayley graph gives a class of graphs that includes many important examples such as the Petersen graph, the Gray graph and the Hoffman-Singleton graph. A bi-Cayley graph Γ over a group H is called normal if H is normal in the full automorphism group of Γ, and almost-normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of almost-normal edge-transitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, ‘almost-normal edge-transitive’ is the same as ‘normal’ for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6. Also we answer some open questions from the literature about these and related classes of graphs, posed by Li (in Proc. Amer. Math. Soc. 133 (2005)), Marušič and Potočnik (in Eur. J. Comb. 22 (2001)) and Marušič and Šparl (in J. Algebr. Comb. 28 (2008)), and we correct a major error in a recent paper by Li, Song and Wang (in J. Comb. Theory Ser. A 120 (2013)).

曲线图Γ承认一组^ h的顶点作用半定期与正好两个轨道构的被称为双Cayley图上^ h。Cayley图的这种概括给出了一类图，其中包括许多重要示例，例如Petersen图，Gray图和Hoffman-Singleton图。在基团A双Cayley图Γ ħ被称为正常如果ħ是正常的完整构群Γ的，并且几乎正常边传递如果满足规范器ħ完全自同构群中的Γ在Γ的边缘上是传递的。在本文中，我们给出了几乎正常的边缘传递双凯伊图的特征，特别是，我们对2弧传递的正常双凯伊图进行了详细描述。利用这一点，我们研究了三类双Cayley图，即那些在阿贝尔群，二面体群和元环p-群上的图。我们发现，在某些条件下，这三类图中的“几乎法线边缘传递”与“法线”相同。作为副产品，我们获得了至多6个周长的所有连接的三价边缘传递图的完整分类。此外，我们还回答了文献中有关这些图和相关类的一些未解决的问题，这些问题由Li（Proc。Amer）提出。数学Soc133（2005）），马鲁希奇和波托奇尼克（以欧元。J.梳子。 22（2001））和马鲁希奇和Šparl（在J. Algebr。梳子。 28（2008）），我们纠正最近的一篇文章一大错误由Li，Song和Wang（在J. Comb。Theory Ser.A 120（2013）中发表）。