209 research outputs found
House of Graphs: a database of interesting graphs
In this note we present House of Graphs (http://hog.grinvin.org) which is a
new database of graphs. The key principle is to have a searchable database and
offer -- next to complete lists of some graph classes -- also a list of special
graphs that already turned out to be interesting and relevant in the study of
graph theoretic problems or as counterexamples to conjectures. This list can be
extended by users of the database.Comment: 8 pages; added a figur
Intertwining wavelets or Multiresolution analysis on graphs through random forests
We propose a new method for performing multiscale analysis of functions
defined on the vertices of a finite connected weighted graph. Our approach
relies on a random spanning forest to downsample the set of vertices, and on
approximate solutions of Markov intertwining relation to provide a subgraph
structure and a filter bank leading to a wavelet basis of the set of functions.
Our construction involves two parameters q and q'. The first one controls the
mean number of kept vertices in the downsampling, while the second one is a
tuning parameter between space localization and frequency localization. We
provide an explicit reconstruction formula, bounds on the reconstruction
operator norm and on the error in the intertwining relation, and a Jackson-like
inequality. These bounds lead to recommend a way to choose the parameters q and
q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure
The Jungle Universe
In this paper, we exploit the fact that the dynamics of homogeneous and
isotropic Friedmann-Lemaitre universes is a special case of generalized
Lotka-Volterra system where the competitive species are the barotropic fluids
filling the Universe. Without coupling between those fluids, Lotka-Volterra
formulation offers a pedagogical and simple way to interpret usual
Friedmann-Lemaitre cosmological dynamics. A natural and physical coupling
between cosmological fluids is proposed which preserve the structure of the
dynamical equations. Using the standard tools of Lotka-Volterra dynamics, we
obtain the general Lyapunov function of the system when one of the fluids is
coupled to dark energy. This provides in a rigorous form a generic asymptotic
behavior for cosmic expansion in presence of coupled species, beyond the
standard de Sitter, Einstein-de Sitter and Milne cosmologies. Finally, we
conjecture that chaos can appear for at least four interacting fluids.Comment: 26 pages, 4 figure
Exponential moments of self-intersection local times of stable random walks in subcritical dimensions
Let be an -stable random walk with values in
. Let be its local time. For ,
not necessarily integer, is the so-called -fold
self- intersection local time of the random walk. When , we
derive precise logarithmic asymptotics of the probability for
all scales r_t \gg \E(I_t). Our result extends previous works by Chen, Li and
Rosen 2005, Becker and K\"onig 2010, and Laurent 2012
Using Graph Theory to Derive Inequalities for the Bell Numbers
The Bell numbers count the number of different ways to partition a set of
elements while the graphical Bell numbers count the number of non-equivalent
partitions of the vertex set of a graph into stable sets. This relation between
graph theory and integer sequences has motivated us to study properties on the
average number of colors in the non-equivalent colorings of a graph to discover
new non trivial inequalities for the Bell numbers. Example are given to
illustrate our approach
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