Perfect state transfer of Grover walks on unitary Cayley graphs

The unitary Cayley graph has vertex set ₙ =\0, 1, , n-1\, where two vertices u and v are adjacent if gcd (u - v, n) = 1. We focus on the periodicity and perfect state transfer of Grover walk on the unitary Cayley graphs. We completely characterize which unitary Cayley graphs are periodic. From our results, we find infinitely many new periodic graphs. We prove that periodicity is a necessary condition for perfect state transfer on an integral vertex-transitive graph, and we provide a simple criterion to characterize perfect state transfer on circulant graphs in terms of its adjacency spectrum. Using these, we prove only three graphs in a class of unitary Cayley graphs exhibit perfect state transfer. Also, we provide a spectral characterization of the periodicity of Grover walk on any integral regular graphs.