Combinatorial necessary conditions for regular graphs to induce periodic quantum walks

We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. If the walk is periodic, all eigenvalues of time evolution matrices must algebraic integers. Focusing on this, we explore which ring coefficients characteristic polynomials should belong to. On other hand, $\eta$-Hermitian adjacency have implications. From these, can find implications in matrices, and thus For example, if a $k$-regular graph with $n$ vertices then $2n/k$ an integer. As application this work, determine periodicity complete prime number vertices.