  The adjacency matrix of a graph G is the Hamiltonian for a continuous-time
quantum walk on the vertices of G. Although the entries of the adjacency matrix
are integers, its eigenvalues are generally irrational and, because of this,
the behaviour of the walk is typically not periodic. In consequence we can
usually only compute numerical approximations to parameters of the walk. In
this paper, we develop theory to exactly study any quantum walk generated by an
integral Hamiltonian. As a result, we provide exact methods to compute the
average of the mixing matrices, and to decide whether pretty good (or almost)
perfect state transfer occurs in a given graph. We also use our methods to
study geometric properties of beautiful curves arising from entries of the
quantum walk matrix, and discuss possible applications of these results.
