Symmetry of graphs and perfect state transfer in Grover walks

We study relationships between symmetry of graphs and perfect state transfer in Grover walks. Symmetry of graphs mathematically refers to automorphisms of graphs. When perfect state transfer occurs between two vertices, the following two statements hold. One is that automorphisms preserve the occurrence of perfect state transfer. The other is that the stabilizer subgroups of the automorphism groups with respect to those two vertices coincide. Using these results, we completely characterize circulant graphs up to valency 4 that admit perfect state transfer. Its proof uses also algebraic number theory.