Further results on the perfect state transfer in integral circulant graphs

For a given graph G, denote by A its adjacency matrix and F (t) = exp(iAt). We say that there exist a perfect state transfer (PST) in G if |F (τ )ab| = 1, for some vertices a, b and a positive real number τ . Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant graphs (circulant graphs with integer eigenvalues). Some results on this topic have already been obtained by Saxena et al. (2007) [5], Bašić et al. (2009) [6] and Basić & Petković (2009) [7]. In this paper, we show that there exists an integral circulant graph with n vertices having a perfect state transfer if and only if 4 | n. Several classes of integral circulant graphs have been found that have a perfect state transfer for the values of n divisible by 4. Moreover, we prove the nonexistence of a PST for several other classes of integral circulant graphs whose order is divisible by 4. These classes cover the class of graphs where the divisor set contains exactly two elements. The obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer.