716 research outputs found
Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems
Inspired by some intriguing examples, we study uniform association schemes
and uniform coherent configurations, including cometric Q-antipodal association
schemes. After a review of imprimitivity, we show that an imprimitive
association scheme is uniform if and only if it is dismantlable, and we cast
these schemes in the broader context of certain --- uniform --- coherent
configurations. We also give a third characterization of uniform schemes in
terms of the Krein parameters, and derive information on the primitive
idempotents of such a scheme. In the second half of the paper, we apply these
results to cometric association schemes. We show that each such scheme is
uniform if and only if it is Q-antipodal, and derive results on the parameters
of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We
revisit the correspondence between uniform indecomposable three-class schemes
and linked systems of symmetric designs, and show that these are cometric
Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class
schemes in terms of only a few parameters, and show that any strongly regular
graph with a ("non-exceptional") strongly regular decomposition gives rise to
such a scheme. Hemisystems in generalized quadrangles provide interesting
examples of such decompositions. We finish with a short discussion of
five-class schemes as well as a list of all feasible parameter sets for
cometric Q-antipodal four-class schemes with at most six fibres and fibre size
at most 2000, and describe the known examples. Most of these examples are
related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions,
April 201
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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