Hostname: page-component-7dd5485656-j7khd Total loading time: 0 Render date: 2025-10-30T14:05:12.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Lower Bounds for Partial Matchings in Regular Bipartite Graphs and Applications to the Monomer–Dimer Entropy

Published online by Cambridge University Press:  01 May 2008

SHMUEL FRIEDLAND  and
LEONID GURVITS
SHMUEL FRIEDLAND
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045, USA (e-mail: friedlan@uic.edu)
LEONID GURVITS
Affiliation:
Los Alamos National Laboratories, Los Alamos, NM 87545, USA (e-mail: gurvits@lanl.gov)
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive here the Friedland–Tverberg inequality for positive hyperbolic polynomials. This inequality is applied to give lower bounds for the number of matchings in r-regular bipartite graphs. It is shown that some of these bounds are asymptotically sharp. We improve the known lower bound for the three-dimensional monomer–dimer entropy.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 3 , May 2008 , pp. 347 - 361
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Baxter, R. J. (1968) Dimers on a rectangular lattice. J. Math. Phys. 9 650–654.CrossRefGoogle Scholar
[2]Bregman, L. M. (1973) Some properties of nonnegative matrices and their permanents. Soviet Math. Dokl. 14 945–949.Google Scholar
[3]Egorichev, G. P. (1981) Proof of the van der Waerden conjecture for permanents. Siberian Math. J. 22 854–859.Google Scholar
[4]Erdős, P. and Rényi, A. (1968) On random matrices II. Studia Math. Hungar. 3 459–464.Google Scholar
[5]Falikman, D. I. (1981) Proof of the van der Waerden conjecture regarding the permanent of doubly stochastic matrix. Math. Notes Acad. Sci. USSR 29 475–479.Google Scholar
[6]Friedland, S. (1979) A lower bound for the permanent of doubly stochastic matrices. Ann. of Math. 110 167–176.CrossRefGoogle Scholar
[7]Friedland, S. (1982) A proof of a generalized van der Waerden conjecture on permanents. Linear Multilinear Algebra 11 107–120.CrossRefGoogle Scholar
[8]Friedland, S. and Gurvits, L. (2006) Generalized Friedland-Tverberg inequality: Applications and extensions. arXiv:math/0603410 v2.Google Scholar
[9]Friedland, S., Krop, E., Lundow, P. H. and Markström, K. (2007) Validations of the asymptotic matching conjectures. arXiv:math.CO/0603001 v1.Google Scholar
[10]Friedland, S., Krop, E. and Markström, K. Enumerating matchings in regular graphs, in preparation.Google Scholar
[11]Friedland, S. and Peled, U. N. (2005) Theory of computation of multidimensional entropy with an application to the monomer-dimer problem. Adv. Appl. Math. 34 486–522.CrossRefGoogle Scholar
[12]Friedland, S. and Peled, U. N. The pressure associated with multidimensional SOFT. In preparation.Google Scholar
[13]Gurvits, L. (2004) Combinatorial and algorithmic aspects of hyperbolic polynomials. arXIv: math.CO/0404474.Google Scholar
[14]Gurvits, L. (2006) Hyperbolic polynomials approach to van der Waerden/Schrijver-Valiant like conjectures: Sharper bounds, simpler proofs and algorithmic applications. In STOC'06: Proc. 38th Annual ACM Symposium on Theory of Computing, ACM, New York, pp. 417–426. arXIv: math.CO/0510452 (2005).CrossRefGoogle Scholar
[15]Hammersley, J. M. (1966) Existence theorems and Monte Carlo methods for the monomer-dimer problem. In Research Papers in Statistics: Festschrift for J. Neyman (David, F. N., ed.), Wiley, London, pp. 125–146.Google Scholar
[16]Heilmann, O. J. and Lieb, E. H. (1972) Theory of monomer-dimer systems. Comm. Math. Phys. 25 190–232.CrossRefGoogle Scholar
[17]Lovász, L. and Plummer, M. D. (1986) Matching Theory, Vol. 121 of North-Holland Mathematical Studies, North-Holland, Amsterdam.Google Scholar
[18]Schrijver, A. (1998) Counting 1-factors in regular bipartite graphs. J. Combin. Theory Ser. B 72 122–135.CrossRefGoogle Scholar
[19]Schrijver, A. and Valiant, W. G. (1980) On lower bounds for permanents. Indagationes Mathematicae 42 425–427.CrossRefGoogle Scholar
[20]Sinkhorn, R. (1964) A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann. Math. Statist. 35 876–879.CrossRefGoogle Scholar
[21]Tverberg, H. (1963) On the permanent of bistochastic matrix. Math. Scand. 12 25–35.CrossRefGoogle Scholar
[22]van der Waerden, B. L. (1926) Aufgabe 45. Jahresber. Deutsch. Math.-Verein. 35 117.Google Scholar
[23]Wanless, I. M. (2003) A lower bound on the maximum permanent in Λkn. Linear Algebra Appl. 373 153–167.CrossRefGoogle Scholar