  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 $t_{G}\Spec(G)
\subseteq \ZZ\pi$, 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
$t_{H}|V_{H}|\Spec(G) \subseteq 2\ZZ\pi$, 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 $K_{1}+\GG\circ\GG+K_{1}$ 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.
