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 mentioned 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 integral circulant graphs with minimal vertices allowing perfect state transfer, other than unitary Cayley graphs, are ICG8({1, 2}) and ICG8({1, 4}). Moreover, it is also conjectured that two classes of integral circulant graphs ICGn({1, n/4}) and ICGn({1, n/2}) allow PST where n ∈ 8N. These conjectures are confirmed in this paper. 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 square-free integer. AMS Subj. Class.: 05C12, 05C50