When can perfect state transfer occur?

Let X be a graph on n vertices with adjacency matrix A and let H(t) denote the matrix-valued function exp(iAt). If u and v are distinct vertices in X, we say perfect state transfer from u to v occurs if there is a time τ such that |H(τ)u,v| = 1. The chief problem is to characterize the cases where perfect state transfer occurs. In this paper, it is shown that if perfect state transfer does occur in a graph, then the square of its spectral radius is either an integer or lies in a quadratic extension of the rationals. From this it is deduced that for any integer k there only finitely many graphs with maximum valency k on which perfect state transfer occurs. It is also shown that if perfect state transfer from u to v occurs, then the graphs X \u and X \ v are cospectral and any automorphism of X that fixes u must fix v (and conversely).