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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
is a graph with perfect state transfer at time , where t_{G}\Spec(G)
\subseteq \ZZ\pi, and is a circulant with odd eigenvalues, their weak
product has perfect state transfer. Also, if is a regular
graph with perfect state transfer at time and is a graph where
t_{H}|V_{H}|\Spec(G) \subseteq 2\ZZ\pi, their lexicographic product
has perfect state transfer. (2) The double cone on any
connected graph , has perfect state transfer if the weights of the cone
edges are proportional to the Perron eigenvector of . 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
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