  The intention of the paper is to move a step towards a classification of
network topologies that exhibit periodic quantum dynamics. We show that the
evolution of a quantum system, whose hamiltonian is identical to the adjacency
matrix of a circulant graph, is periodic if and only if all eigenvalues of the
graph are integers (that is, the graph is integral). Motivated by this
observation, we focus on relevant properties of integral circulant graphs.
Specifically, we bound the number of vertices of integral circulant graphs in
terms of their degree, characterize bipartiteness and give exact bounds for
their diameter. Additionally, we prove that circulant graphs with odd order do
not allow perfect state transfer.
