43,664 research outputs found
On a decomposition of regular domains into John domains with uniform constants
We derive a decomposition result for regular, two-dimensional domains into
John domains with uniform constants. We prove that for every simply connected
domain with -boundary there is a corresponding
partition with such
that each component is a John domain with a John constant only depending on
. The result implies that many inequalities in Sobolev spaces such as
Poincar\'e's or Korn's inequality hold on the partition of for uniform
constants, which are independent of
Spectral properties of random graphs with fixed equitable partition
We define a graph to be -regular if it contains an equitable partition
given by a matrix . These graphs are generalizations of both regular and
bipartite, biregular graphs. An -regular matrix is defined then as a matrix
on an -regular graph consistent with the graph's equitable partition. In
this paper we derive the limiting spectral density for large, random
-regular matrices as well as limiting functions of certain statistics for
their eigenvector coordinates as a function of eigenvalue. These limiting
functions are defined in terms of spectral measures on -regular trees. In
general, these spectral measures do not have a closed-form expression; however,
we provide a defining system of polynomials for them. Finally, we explore
eigenvalue bounds of -regular graph, proving an expander mixing lemma,
Alon-Bopana bound, and other eigenvalue inequalities in terms of the
eigenvalues of the matrix .Comment: 24 pages, 3 figure
Rado functionals and applications
We study Rado functionals and the maximal condition (first introduced by J.
M. Barret et al.) in terms of the partition regularity of mixed systems of
linear equations and inequalities. By strengthening the maximal Rado condition,
we provide a sufficient condition for the partition regularity of polynomial
equations over some infinite subsets of a given commutative ring. By applying
these results, we derive an extension of a previous result obtained by M. Di
Nasso and L. Luperi Baglini concerning partition regular inhomogeneous
polynomials in three variables and also conditions for the partition regularity
of equations of the form , where is a non-zero rational
and is a homogeneous polynomial
A reverse Sidorenko inequality
Let be a graph allowing loops as well as vertex and edge weights. We
prove that, for every triangle-free graph without isolated vertices, the
weighted number of graph homomorphisms satisfies the inequality
where denotes the degree of vertex in . In particular, one has for every -regular
triangle-free . The triangle-free hypothesis on is best possible. More
generally, we prove a graphical Brascamp-Lieb type inequality, where every edge
of is assigned some two-variable function. These inequalities imply tight
upper bounds on the partition function of various statistical models such as
the Ising and Potts models, which includes independent sets and graph
colorings.
For graph colorings, corresponding to , we show that the
triangle-free hypothesis on may be dropped; this is also valid if some of
the vertices of are looped. A corollary is that among -regular graphs,
maximizes the quantity for every and ,
where counts proper -colorings of .
Finally, we show that if the edge-weight matrix of is positive
semidefinite, then This implies that among -regular graphs,
maximizes . For 2-spin Ising models, our results give a
complete characterization of extremal graphs: complete bipartite graphs
maximize the partition function of 2-spin antiferromagnetic models and cliques
maximize the partition function of ferromagnetic models.
These results settle a number of conjectures by Galvin-Tetali, Galvin, and
Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture
by Kahn.Comment: 30 page
New bounds for the max--cut and chromatic number of a graph
We consider several semidefinite programming relaxations for the max--cut
problem, with increasing complexity. The optimal solution of the weakest
presented semidefinite programming relaxation has a closed form expression that
includes the largest Laplacian eigenvalue of the graph under consideration.
This is the first known eigenvalue bound for the max--cut when that is
applicable to any graph. This bound is exploited to derive a new eigenvalue
bound on the chromatic number of a graph. For regular graphs, the new bound on
the chromatic number is the same as the well-known Hoffman bound; however, the
two bounds are incomparable in general. We prove that the eigenvalue bound for
the max--cut is tight for several classes of graphs. We investigate the
presented bounds for specific classes of graphs, such as walk-regular graphs,
strongly regular graphs, and graphs from the Hamming association scheme
Semidefinite programming and eigenvalue bounds for the graph partition problem
The graph partition problem is the problem of partitioning the vertex set of
a graph into a fixed number of sets of given sizes such that the sum of weights
of edges joining different sets is optimized. In this paper we simplify a known
matrix-lifting semidefinite programming relaxation of the graph partition
problem for several classes of graphs and also show how to aggregate additional
triangle and independent set constraints for graphs with symmetry. We present
an eigenvalue bound for the graph partition problem of a strongly regular
graph, extending a similar result for the equipartition problem. We also derive
a linear programming bound of the graph partition problem for certain Johnson
and Kneser graphs. Using what we call the Laplacian algebra of a graph, we
derive an eigenvalue bound for the graph partition problem that is the first
known closed form bound that is applicable to any graph, thereby extending a
well-known result in spectral graph theory. Finally, we strengthen a known
semidefinite programming relaxation of a specific quadratic assignment problem
and the above-mentioned matrix-lifting semidefinite programming relaxation by
adding two constraints that correspond to assigning two vertices of the graph
to different parts of the partition. This strengthening performs well on highly
symmetric graphs when other relaxations provide weak or trivial bounds
Remarks on the boundary set of spectral equipartitions
Given a bounded open set in (or a compact Riemannian
manifold with boundary), and a partition of by open sets
, we consider the quantity , where
is the ground state energy of the Dirichlet realization of
the Laplacian in . We denote by the infimum
of over all -partitions. A minimal -partition
is a partition which realizes the infimum. The purpose of this paper is to
revisit properties of nodal sets and to explore if they are also true for
minimal partitions, or more generally for spectral equipartitions. We focus on
the length of the boundary set of the partition in the 2-dimensional situation.Comment: Final version to appear in the Philosophical Transactions of the
Royal Society
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