Periodicities of Grover Walks on Distance-Regular Graphs

Characterization of some classes of graphs inducing periodic Grover walks have been studied for recent years. In particular, for the strongly regular graphs, it has been known that there are only three kinds of such graphs. As an extension of this result, we focus on the periodicity of the Grover walks on distance-regular graphs. The distance-regular graph is regarded as a kind of generalization of the strongly regular graph. In this paper, we find some classes of distance-regular graphs admitting a periodic Grover walk and obtain a useful necessary condition to induce periodic Grover walks. Also, we apply this necessary condition to give another proof for the classification of the strongly regular graphs.