Perfect state transfer on quasi-abelian semi-Cayley graphs

Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. A graph is a semi-Cayley graph over a group G if it admits G as a semiregular subgroup of the full automorphism group with two orbits of equal size. A semi-Cayley graph SC(G, R, L, S) is called quasi-abelian if each of R, L and S is a union of some conjugacy classes of G. This paper establishes necessary and sufficient conditions for a quasi-abelian semi-Cayley graph to have perfect state transfer. As a corollary, it is shown that if a quasi-abelian semi-Cayley graph over a finite group G has perfect state transfer between distinct vertices g and h, and G has a faithful irreducible character, then \(gh^{-1}\) lies in the center of G and \(gh=hg\); in particular, G cannot be a non-abelian simple group. We also characterize quasi-abelian Cayley graphs over arbitrary groups having perfect state transfer, which is a generalization of previous works on Cayley graphs over abelian groups, dihedral groups, semi-dihedral groups and dicyclic groups.