Some results for the periodicity and perfect state transfer (Q640442)

Summary: 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 \(\tau \) such that \(H(\tau )\) 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 \(\tau \) such that \(|H(\tau )_{u,v}| = 1\). A necessary and sufficient condition for \(G\) to be periodic is given. We show the existence of perfect state transfer between antipodal vertices in graphs with extreme diameter.