Some classes of integral circulant graphs either allowing or not allowing perfect state transfer

The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. Motivated by the aforementioned work, Bašić, Petković and Stevanović give the simple condition for the characterization of integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. They stated that the integral circulant graphs with minimal vertices allowing perfect state transfer, other than unitary Cayley graphs, are ICG 8 ( { 1 , 2 } ) and ICG 8 ( { 1 , 4 } ) . Moreover, it is also conjectured that two classes of integral circulant graphs ICG n ( { 1 , n / 4 } ) and ICG n ( { 1 , n / 2 } ) allow PST where n ∈ 8 N . These conjectures are confirmed in this work. Moreover, it is shown that there are no integral circulant graphs allowing perfect state transfer in the class of graphs where the number of vertices is a square-free integer.