In this paper, we determine periodicity of quantum walks defined by mixed paths and mixed cycles. By the spectral mapping theorem of quantum walks, consideration of periodicity is reduced to eigenvalue analysis of η-Hermitian adjacency matrices. First, we investigate coefficients of the characteristic polynomials of η-Hermitian adjacency matrices. We show that the characteristic polynomials of mixed trees and their underlying graphs are same. We also define $n+1$ types of mixed cycles and show that every mixed cycle is switching equivalent to one of them. We use these results to discuss periodicity. We show that the mixed paths are periodic for any η. In addition, we provide a necessary and sufficient condition for a mixed cycle to be periodic and determine their periods.

在本文中，我们确定了由混合路径和混合循环定义的量子行走的周期性。通过量子游走的谱映射定理，将周期性的考虑简化为η- Hermitian邻接矩阵的特征值分析。首先，我们研究了η -Hermitian 邻接矩阵的特征多项式的系数。我们证明了混合树的特征多项式及其底层图是相同的。我们还定义$n+1$混合循环的类型，并表明每个混合循环都切换等价于其中之一。我们使用这些结果来讨论周期性。我们证明混合路径对于任何η都是周期性的。此外，我们提供了混合循环具有周期性的充要条件，并确定了它们的周期。