  Quantum walks determined by the coin operator on graphs have been intensively
studied. The typical examples of coin operator are the Grover and Fourier
matrices. The periodicity of the Grover walk is well investigated. However, the
corresponding result on the Fourier walk is not known. In this paper, we
present a necessary condition for the Fourier walk on regular graphs to have
the finite period. As an application of our result, we show that the Fourier
walks do not have any finite period for some classes of regular graphs such as
complete graphs, cycle graphs with selfloops, and hypercubes.
