64,598 research outputs found
Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups
In this paper, we present a method to obtain regular (or equitable)
partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of
permutation groups on letters. We prove that every partition of the number
gives rise to a regular partition of the Cayley graph. By using
representation theory, we also obtain the complete spectra and the eigenspaces
of the corresponding quotient (di)graphs. More precisely, we provide a method
to find all the eigenvalues and eigenvectors of such (di)graphs, based on their
irreducible representations. As examples, we apply this method to the pancake
graphs and to a recent known family of mixed graphs
(having edges with and without direction). As a byproduct, the existence of
perfect codes in allows us to give a lower bound for the multiplicity of
its eigenvalue
Double-Edge Factor Graphs: Definition, Properties, and Examples
Some of the most interesting quantities associated with a factor graph are
its marginals and its partition sum. For factor graphs \emph{without cycles}
and moderate message update complexities, the sum-product algorithm (SPA) can
be used to efficiently compute these quantities exactly. Moreover, for various
classes of factor graphs \emph{with cycles}, the SPA has been successfully
applied to efficiently compute good approximations to these quantities. Note
that in the case of factor graphs with cycles, the local functions are usually
non-negative real-valued functions. In this paper we introduce a class of
factor graphs, called double-edge factor graphs (DE-FGs), which allow local
functions to be complex-valued and only require them, in some suitable sense,
to be positive semi-definite. We discuss various properties of the SPA when
running it on DE-FGs and we show promising numerical results for various
example DE-FGs, some of which have connections to quantum information
processing.Comment: Submitte
Right-convergence of sparse random graphs
The paper is devoted to the problem of establishing right-convergence of
sparse random graphs. This concerns the convergence of the logarithm of number
of homomorphisms from graphs or hyper-graphs \G_N, N\ge 1 to some target
graph . The theory of dense graph convergence, including random dense
graphs, is now well understood, but its counterpart for sparse random graphs
presents some fundamental difficulties. Phrased in the statistical physics
terminology, the issue is the existence of the log-partition function limits,
also known as free energy limits, appropriately normalized for the Gibbs
distribution associated with . In this paper we prove that the sequence of
sparse \ER graphs is right-converging when the tensor product associated with
the target graph satisfies certain convexity property. We treat the case of
discrete and continuous target graphs . The latter case allows us to prove a
special case of Talagrand's recent conjecture (more accurately stated as level
III Research Problem 6.7.2 in his recent book), concerning the existence of the
limit of the measure of a set obtained from by intersecting it with
linearly in many subsets, generated according to some common probability
law.
Our proof is based on the interpolation technique, introduced first by Guerra
and Toninelli and developed further in a series of papers. Specifically, Bayati
et al establish the right-convergence property for Erdos-Renyi graphs for some
special cases of . In this paper most of the results in this paper follow as
a special case of our main theorem.Comment: 22 page
Conformal Field Theories, Graphs and Quantum Algebras
This article reviews some recent progress in our understanding of the
structure of Rational Conformal Field Theories, based on ideas that originate
for a large part in the work of A. Ocneanu. The consistency conditions that
generalize modular invariance for a given RCFT in the presence of various types
of boundary conditions --open, twisted-- are encoded in a system of integer
multiplicities that form matrix representations of fusion-like algebras. These
multiplicities are also the combinatorial data that enable one to construct an
abstract ``quantum'' algebra, whose - and -symbols contain essential
information on the Operator Product Algebra of the RCFT and are part of a cell
system, subject to pentagonal identities. It looks quite plausible that the
classification of a wide class of RCFT amounts to a classification of ``Weak
- Hopf algebras''.Comment: 23 pages, 12 figures, LateX. To appear in MATHPHYS ODYSSEY 2001
--Integrable Models and Beyond, ed. M. Kashiwara and T. Miwa, Progress in
Math., Birkhauser. References and comments adde
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