Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T22:16:27.212Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact and Approximate Compression of Transfer Matrices for Graph Homomorphisms

Published online by Cambridge University Press:  01 February 2010

Per Håkan Lundow  and
Klas Markström
Per Håkan Lundow
Affiliation:
KTH Physics, AlbaNova University Center, SE-106 91 Stockholm, Sweden, phl@kth.se, http://www.theophys.kth.se/~phl
Klas Markström
Affiliation:
Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden, Klas.Markstrom@math.umu.se, http://abel.math.umu.se/~klasm

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The aim of this paper is to extend the previous work on transfer matrix compression in the case of graph homomorphisms. For H-homomorphisms of lattice-like graphs we demonstrate how the automorphisms of H, as well as those of the underlying lattice, can be used to reduce the size of the relevant transfer matrices. As applications of this method we give currently best known bounds for the number of 4- and 5-colourings of the square grid, and the number of 3- and 4-colourings of the three-dimensional cubic lattice. Finally, we also discuss approximate compression of transfer matrices.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 11 , 2008 , pp. 1 - 14
Copyright
Copyright © London Mathematical Society 2008

References

1.Babić, Darko, Graovac, Ante, Mohar, Bojan and Pisanski, Tomaž, ‘The matching polynomial of a polygraph’, Discrete Appl. Math. 15 (1986) 11–24.CrossRefGoogle Scholar
2.Bakaev, A. V. and Kabanovich, V. I., ‘Series expansions for the q-colour problem on the square and cubic lattices’, J. Phys. A 27 (1994) 6731–6739.Google Scholar
3.Brightwell, Graham R. and Winkler, Peter, ‘Graph homomorphisms and phase transitions’, J. Cornbin. Theory Ser. B 77 (1999) 221–262.CrossRefGoogle Scholar
4.Chang, Shu-Chiuan and Shrock, Robert, ‘Tutte polynomials and related asymptotic limiting functions for recursive families of graphs’, Adv. in Appl. Math. 32, Special issue on the Tutte polynomial (2004) 44–87.CrossRefGoogle Scholar
5.Ciucu, M., ‘An improved upper bound for the 3-dimensional dimer problem’, Duke Math. J. 94 (1998) 1–11.CrossRefGoogle Scholar
6.Freedman, M., Lovász, L. and Schrijver, A., ‘Reflection positivity, rank connectivity, and homomorphism of graphs’, J. Amer. Math. Soc. 20 (2007) 37–51.CrossRefGoogle Scholar
7.Friedland, Shmuel and Peled, Uri N., ‘Theory of computation of multidimensional entropy with an application to the monomer-dimer problem’, Adv. in Appl. Math. 34 (2005) 486–522.CrossRefGoogle Scholar
8.Godsil, Chris and Royle, Gordon, Algebraic graph theory, Graduate Texts in Mathematics 207 (Springer, New York, 2001).CrossRefGoogle Scholar
9.Häggkvist, R. and Lundow, P. H., ‘The Ising partition function for 2D grids with cyclic boundary: computation and analysis’, J. Statist. Phys. 108 (2002) 429–457.CrossRefGoogle Scholar
10.Jacobsen, Jesper Lykke and Salas, Jesús, ‘Transfer matrices and partition-function zeros for antiferromagnetic Potts models. II. Extended results for square-lattice chromatic polynomial’, J. Statist. Phys. 104 (2001) 701–723.CrossRefGoogle Scholar
11.Jacobsen, Jesper Lykke, Salas, Jesús and Sokal, Alan D., ‘Transfer matrices and partition-function zeros for antiferromagnetic Potts models. III. Triangular-lattice chromatic polynomial’, J. Statist. Phys. 112 (2003) 921–1017.CrossRefGoogle Scholar
12.Lieb, Elliott H., ‘Residual entropy of square ice’, Phys. Rev. 162 (1967) 162–172.CrossRefGoogle Scholar
13.Lundow, Per Håkan, ‘Compression of transfer matrices’, 17th British Combinatorial Conference, Canterbury, 1999, Discrete Math. 231 (2001) 321–329.CrossRefGoogle Scholar
14.Read, R. C. and Tutte, W. T., ‘Chromatic polynomials’, Selected topics in graph theory 3 (Academic Press, San Diego, CA, 1988) 15–42.Google Scholar
15.Salas, Jesús and Sokal, Alan D., ‘Transfer matrices and partition-function zeros for antiferromagnetic Potts models. I. General theory and square-lattice chromatic polynomial.’, J. Statist. Phys. 104 (2001) 609–699.CrossRefGoogle Scholar
16.Wu, F. Y., ‘The Potts model’, Rev. Modern Phys. 54 (1982) 235–268.CrossRefGoogle Scholar