  For a quantum walk on a graph, there exist many kinds of operators for the
discrete-time evolution. We give a general relation between the characteristic
polynomial of the evolution matrix of a quantum walk on edges and that of a
kind of transition matrix of a classical random walk on vertices. Furthermore
we determine the structure of the positive support of the cube of some
evolution matrix, which is said to be useful for isospectral problem in graphs,
under a certain condition.
