Perfect state transfer, graph products and equitable partitions

We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If G is a graph with perfect state transfer at time t_G, where t_GSpec(G) subseteq ZZπ, and H is a circulant with odd eigenvalues, their weak product G times H has perfect state transfer. Also, if H is a regular graph with perfect state transfer at time t_H and G is a graph where t_H|V_H|Spec(G) subseteq 2ZZπ, their lexicographic product G[H] has perfect state transfer. (2) The double cone overline{K}₂ + G on any connected graph G, has perfect state transfer if the weights of the cone edges are proportional to the Perron eigenvector of G. This generalizes results for double cone on regular graphs studied in [BCMS09,ANOPRT10,ANOPRT09]. (3) For an infinite family GG of regular graphs, there is a circulant connection so the graph K₁+GGcircGG+K₁ has perfect state transfer. In contrast, no perfect state transfer exists if a complete bipartite connection is used (even in the presence of weights) [ANOPRT09]. We also describe a generalization of the path collapsing argument [CCDFGS03,CDDEKL05], which reduces questions about perfect state transfer to simpler (weighted) multigraphs, for graphs with equitable distance partitions.