  The conventional spectral mapping theorem for quantum walks can only be
applied for walks employing a shift operator whose square is the identity. This
theorem gives most of the eigenvalues of the time evolution $U$ by lifting the
eigenvalues of an induced self-adjoint matrix $T$ onto the unit circle on the
complex plane. We acquire a new spectral mapping theorem for the Grover walk
with a shift operator whose cube is the identity on finite graphs. Moreover,
graphs we can consider for a quantum walk with such a shift operator is
characterized by a triangulation. We call these graphs triangulable graphs in
this paper. One of the differences between our spectral mapping theorem and the
conventional one is that lifting the eigenvalues of $T-1/2$ onto the unit
circle gives most of the eigenvalues of $U$.
