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 tGβ, where t_{G}\Spec(G)
\subseteq \ZZ\pi, and H is a circulant with odd eigenvalues, their weak
product GΓH has perfect state transfer. Also, if H is a regular
graph with perfect state transfer at time tHβ and G is a graph where
t_{H}|V_{H}|\Spec(G) \subseteq 2\ZZ\pi, their lexicographic product G[H]
has perfect state transfer. (2) The double cone K2β+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 K_{1}+\GG\circ\GG+K_{1} 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.Comment: 18 pages, 6 figure