Abstract
A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits of equal size. In this paper, some basic properties and the automorphisms of bi-Cayley graphs are explored. As an application, a classification of connected cubic vertex-transitive bi-Cayley graphs over abelian groups is given, and using this, a problem posed in Zhou and Feng (2012) regarding the Cayley property of a class of graphs is solved.
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