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 ∈ 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. 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.