215 research outputs found
Symmetric Interconnection Networks from Cubic Crystal Lattices
Torus networks of moderate degree have been widely used in the supercomputer
industry. Tori are superb when used for executing applications that require
near-neighbor communications. Nevertheless, they are not so good when dealing
with global communications. Hence, typical 3D implementations have evolved to
5D networks, among other reasons, to reduce network distances. Most of these
big systems are mixed-radix tori which are not the best option for minimizing
distances and efficiently using network resources. This paper is focused on
improving the topological properties of these networks.
By using integral matrices to deal with Cayley graphs over Abelian groups, we
have been able to propose and analyze a family of high-dimensional grid-based
interconnection networks. As they are built over -dimensional grids that
induce a regular tiling of the space, these topologies have been denoted
\textsl{lattice graphs}. We will focus on cubic crystal lattices for modeling
symmetric 3D networks. Other higher dimensional networks can be composed over
these graphs, as illustrated in this research. Easy network partitioning can
also take advantage of this network composition operation. Minimal routing
algorithms are also provided for these new topologies. Finally, some practical
issues such as implementability and preliminary performance evaluations have
been addressed
Ramanujan Coverings of Graphs
Let be a finite connected graph, and let be the spectral radius of
its universal cover. For example, if is -regular then
. We show that for every , there is an -covering
(a.k.a. an -lift) of where all the new eigenvalues are bounded from
above by . It follows that a bipartite Ramanujan graph has a Ramanujan
-covering for every . This generalizes the case due to Marcus,
Spielman and Srivastava (2013).
Every -covering of corresponds to a labeling of the edges of by
elements of the symmetric group . We generalize this notion to labeling
the edges by elements of various groups and present a broader scenario where
Ramanujan coverings are guaranteed to exist.
In particular, this shows the existence of richer families of bipartite
Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava,
a crucial component of our proof is the existence of interlacing families of
polynomials for complex reflection groups. The core argument of this component
is taken from a recent paper of them (2015).
Another important ingredient of our proof is a new generalization of the
matching polynomial of a graph. We define the -th matching polynomial of
to be the average matching polynomial of all -coverings of . We show this
polynomial shares many properties with the original matching polynomial. For
example, it is real rooted with all its roots inside .Comment: 38 pages, 4 figures, journal version (minor changes from previous
arXiv version). Shortened version appeared in STOC 201
The Spectra of Lamplighter Groups and Cayley Machines
We calculate the spectra and spectral measures associated to random walks on
restricted wreath products of finite groups with the infinite cyclic group, by
calculating the Kesten-von Neumann-Serre spectral measures for the random walks
on Schreier graphs of certain groups generated by automata. This generalises
the work of Grigorchuk and Zuk on the lamplighter group. In the process we
characterise when the usual spectral measure for a group generated by automata
coincides with the Kesten-von Neumann-Serre spectral measure.Comment: 36 pages, improved exposition, main results slightly strengthene
Schur-Weyl duality and the heat kernel measure on the unitary group
We establish a convergent power series expansion for the expectation of a
product of traces of powers of a random unitary matrix under the heat kernel
measure. These expectations turn out to be the generating series of certain
paths in the Cayley graph of the symmetric group. We then compute the
asymptotic distribution of a random unitary matrix under the heat kernel
measure on the unitary group U(N) as N tends to infinity, and prove a result of
asymptotic freeness result for independent large unitary matrices, thus
recovering results obtained previously by Xu and Biane. We give an
interpretation of our main expansion in terms of random ramified coverings of a
disk. Our approach is based on the Schur-Weyl duality and we extend some of our
results to the orthogonal and symplectic cases
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