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Periodic Graphs

unpublished

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. If u ∈ V (X) and there is a time σ such that |H(σ) u,u | = 1, we say X is periodic at u with period σ. 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

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