Some Results for the Periodicity and Perfect State Transfer

Let G be a graph with adjacency matrix A, let H(t) = exp(itA). G is called a periodic graph if there exists a time τ such that H(τ) is diagonal. If u and v are distinct vertices in G, we say that perfect state transfer occurs from u to v if there exists a time τ such that |H(τ)u,v| = 1. A necessary and sufficient condition for G is periodic is given. We give the existence for the perfect state transfer between antipodal vertices in graphs with extreme diameter.