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{\bf Chris Godsil}
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{\bf Periodic Graphs}
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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 \textsl{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 $\sigma$ such
that $|H(\sigma)_{u,u}|=1$, we say $X$ is \textsl{periodic at $u$} with
period $\sigma$. It is not difficult to show that if the ratio of
distinct non-zero eigenvalues of $X$ is always rational, then $X$ is
periodic. We show that the converse holds, from which it follows that
a regular graph is periodic if and only if its eigenvalues are
distinct. For a class of graphs $X$ including all vertex-transitive
graphs we prove that, 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.
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