  In this paper we study discrete-time quantum walks on Cayley graphs
corresponding to Dihedral groups, which are graphs with both directed and
undirected edges. We consider the walks with coins that are one-parameter
continuous deformation of the Grover matrix and can be written as linear
combinations of certain permutation matrices. We show that the walks are
periodic only for coins that are permutation or negative of a permutation
matrix. Finally, we investigate the localization property of the walks through
numerical simulations and observe that the walks localize for a wide range of
coins for different sizes of the graphs.
