Periodic Graphs

Understand

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