Perfect state transfer on bi-Cayley graphs over abelian groups

The study of perfect state transfer on graphs has attracted a great deal attention during the past ten years because its applications to quantum information processing and computation. Perfect is understood be rare phenomenon. This paper establishes necessary sufficient conditions for bi-Cayley graph having over any given finite abelian group. As corollaries, many known new results are obtained Cayley groups, (generalized) dihedral semi-dihedral groups generalized quaternion groups. Especially, we give an example connected non-normal group between two distinct vertices, which was thought impossible.