  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
$\eta$-Hermitian adjacency matrices. First, we investigate coefficients of the
characteristic polynomials of $\eta$-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 $\eta$. In
addition, we provide a necessary and sufficient condition for a mixed cycle to
be periodic and determine their periods.
