Periodic graphs

Abstract

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 nonzero 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 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.