  Let $X$ be a graph on $n$ vertices with 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 $\tau$ such that $|H({\tau})_{u,v}| = 1$. Our chief problem
is to characterize the cases where perfect state transfer occurs. We show that
if perfect state transfer does occur in a graph, then the spectral radius is an
integer or a quadratic irrational; using this we prove that there are only
finitely many graphs with perfect state transfer and with maximum valency at
most 4K4. We also show that if perfect state transfer from $u$ to $v$ occurs,
then the graphs $X\setminus u$ and $X\setminus v$ are cospectral and any
automorphism of $X$ that fixes $u$ must fix $v$ (and conversely).
