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