43,664 research outputs found

    On a decomposition of regular domains into John domains with uniform constants

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    We derive a decomposition result for regular, two-dimensional domains into John domains with uniform constants. We prove that for every simply connected domain ΩR2\Omega \subset {\Bbb R}^2 with C1C^1-boundary there is a corresponding partition Ω=Ω1ΩN\Omega = \Omega_1 \cup \ldots \cup \Omega_N with j=1NH1(ΩjΩ)θ\sum_{j=1}^N \mathcal{H}^1(\partial \Omega_j \setminus \partial \Omega) \le \theta such that each component is a John domain with a John constant only depending on θ\theta. The result implies that many inequalities in Sobolev spaces such as Poincar\'e's or Korn's inequality hold on the partition of Ω\Omega for uniform constants, which are independent of Ω\Omega

    Spectral properties of random graphs with fixed equitable partition

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    We define a graph to be SS-regular if it contains an equitable partition given by a matrix SS. These graphs are generalizations of both regular and bipartite, biregular graphs. An SS-regular matrix is defined then as a matrix on an SS-regular graph consistent with the graph's equitable partition. In this paper we derive the limiting spectral density for large, random SS-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 SS-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 SS-regular graph, proving an expander mixing lemma, Alon-Bopana bound, and other eigenvalue inequalities in terms of the eigenvalues of the matrix SS.Comment: 24 pages, 3 figure

    Rado functionals and applications

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    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 H(xzρ,y)=0H(xz^\rho,y)=0, where ρ\rho is a non-zero rational and HZ[x,y]H\in\mathbb{Z}[x,y] is a homogeneous polynomial

    A reverse Sidorenko inequality

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    Let HH be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph GG without isolated vertices, the weighted number of graph homomorphisms hom(G,H)\hom(G, H) satisfies the inequality hom(G,H)uvE(G)hom(Kdu,dv,H)1/(dudv), \hom(G, H ) \le \prod_{uv \in E(G)} \hom(K_{d_u,d_v}, H )^{1/(d_ud_v)}, where dud_u denotes the degree of vertex uu in GG. In particular, one has hom(G,H)1/E(G)hom(Kd,d,H)1/d2 \hom(G, H )^{1/|E(G)|} \le \hom(K_{d,d}, H )^{1/d^2} for every dd-regular triangle-free GG. The triangle-free hypothesis on GG is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of GG 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 H=KqH = K_q, we show that the triangle-free hypothesis on GG may be dropped; this is also valid if some of the vertices of KqK_q are looped. A corollary is that among dd-regular graphs, G=Kd,dG = K_{d,d} maximizes the quantity cq(G)1/V(G)c_q(G)^{1/|V(G)|} for every qq and dd, where cq(G)c_q(G) counts proper qq-colorings of GG. Finally, we show that if the edge-weight matrix of HH is positive semidefinite, then hom(G,H)vV(G)hom(Kdv+1,H)1/(dv+1). \hom(G, H) \le \prod_{v \in V(G)} \hom(K_{d_v+1}, H )^{1/(d_v+1)}. This implies that among dd-regular graphs, G=Kd+1G = K_{d+1} maximizes hom(G,H)1/V(G)\hom(G, H)^{1/|V(G)|}. 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-kk-cut and chromatic number of a graph

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    We consider several semidefinite programming relaxations for the max-kk-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-kk-cut when k>2k>2 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-kk-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

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    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

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    Given a bounded open set Ω\Omega in Rn\mathbb{R}^n (or a compact Riemannian manifold with boundary), and a partition of Ω\Omega by kk open sets ωj\omega_j, we consider the quantity maxjλ(ωj)\max_j \lambda(\omega_j), where λ(ωj)\lambda(\omega_j) is the ground state energy of the Dirichlet realization of the Laplacian in ωj\omega_j. We denote by Lk(Ω)\mathfrak{L}_k(\Omega) the infimum of maxjλ(ωj)\max_j \lambda(\omega_j) over all kk-partitions. A minimal kk-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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