The 3-dimensional cube is the only periodic, connected cubic graph with perfect state transfer

There is perfect state transfer between two vertices of a graph, if a single excitation can travel with fidelity one between the corresponding sites of a spin system modeled by the graph. When the excitation is back at the initial site, for all sites at the same time, the graph is said to be periodic. A graph is cubic if each of its vertices has a neighbourhood of size exactly three. We prove that the 3-dimensional cube is the only periodic, connected cubic graph with perfect state transfer. We conjecture that this is also the only connected cubic graph with perfect state transfer.