  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 $|H(\tau)_{u,v}|=1$. If $u\in V(X)$ and
there is a time $\sg$ such that $|H(\sg)_{u,u}|=1$, we say $X$ is periodic at
$u$ with period $\sg$. We show that if perfect state transfer from $u$ to $v$
occurs at time $\tau$, then $X$ is periodic at both $u$ and $v$ with period
$2\tau$. We extend previous work by showing that a regular graph with at least
four distinct eigenvalues is periodic with respect to some vertex if and only
if its eigenvalues are integers. We show that, for a class of graphs $X$
including all vertex-transitive graphs, if perfect state transfer occurs at
time $\tau$, then $H(\tau)$ is a scalar multiple of a permutation matrix of
order two with no fixed points. Using certain Hadamard matrices, we construct a
new infinite family of graphs on which perfect state transfer occurs.
