478 research outputs found
Integral Cayley graphs and groups
We solve two open problems regarding the classification of certain classes of
Cayley graphs with integer eigenvalues. We first classify all finite groups
that have a "non-trivial" Cayley graph with integer eigenvalues, thus solving a
problem proposed by Abdollahi and Jazaeri. The notion of Cayley integral groups
was introduced by Klotz and Sander. These are groups for which every Cayley
graph has only integer eigenvalues. In the second part of the paper, all Cayley
integral groups are determined.Comment: Submitted June 18 to SIAM J. Discrete Mat
Chirality from quantum walks without quantum coin
Quantum walks (QWs) describe the evolution of quantum systems on graphs. An
intrinsic degree of freedom---called the coin and represented by a
finite-dimensional Hilbert space---is associated to each node. Scalar quantum
walks are QWs with a one-dimensional coin. We propose a general strategy
allowing one to construct scalar QWs on a broad variety of graphs, which admit
embedding in Eulidean spaces, thus having a direct geometric interpretation.
After reviewing the technique that allows one to regroup cells of nodes into
new nodes, transforming finite spatial blocks into internal degrees of freedom,
we prove that no QW with a two-dimensional coin can be derived from an
isotropic scalar QW in this way. Finally we show that the Weyl and Dirac QWs
can be derived from scalar QWs in spaces of dimension up to three, via our
construction.Comment: 22 pages, 2 figure
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