Perfect state transfer in Laplacian quantum walk

For a graph G and a related symmetric matrix M, the continuous-time quantum walk on G relative to M is defined as the unitary matrix U(t) = exp(-it M), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time t if the (u, v)-entry of U(tau) has unit magnitude. This paper studies quantumwalks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer. If an n-vertex graph has perfect state transfer at time tau relative to the Laplacian, then so does its complement if n tau is an element of 2 pi Z. As a corollary, the join of (k) over bar (2) with any m-vertex graph has perfect state transfer relative to the Laplacian if and only if m equivalent to 2 (mod 4). This was previously known for the join of (k) over bar (2) with a clique (Bose et al. in Int J Quant Inf 7:713-723, 2009). If a graph G has perfect state transfer at time (tau) relative to the normalized Laplacian, then so does the weak product G x H if for any normalized Laplacian eigenvalues lambda of G and mu of H, we have mu(lambda-1)tau is an element of 2 pi Z. As a corollary, a weak product of P-3 with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of P-3 has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (Godsil in Discret Math 312(1):129-147, 2011).