Which weighted circulant networks have perfect state transfer?

Abstract: The question of perfect state transfer existence in quantum spin networks based on weighted graphs has been recently presented by many authors. We give a simple condition for characterizing weighted circulant graphs allowing perfect state transfer in terms of their eigenvalues. This is done by extending the results about quantum periodicity existence in the networks obtained by Saxena, Severini and Shparlinski and characterizing integral graphs among weighted circulant graphs. Finally, classes of weighted circulant graphs supporting perfect state transfer are found. These classes completely cover the class of circulant graphs having perfect state transfer in the unweighted case. In fact, we show that there exists an weighted integral circulant graph with $n$ vertices having perfect state transfer if and only if $n$ is even. Moreover we prove the non-existence of perfect state transfer for several other classes of weighted integral circulant graphs of even order.