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Periodic Graphs

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 $τ$ such that $|H(τ)_{u,v}|=1$. 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 $τ$, then $X$ is periodic at both $u$ and $v$ with period $2τ$. 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 $τ$, then $H(τ)$ 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.