Ground state entropy of Potts antiferromagnets on homeomorphic families of strip graphs

Physica A: Statistical Mechanics and its Applications

Volumn 259, Issue 3-4, 1998, Pages 315-348

  Shrock, Robert ^a   Tsai, Shan Ho ^a  

^a STONY BROOK UNIVERSITY   (United States)

Author keywords

Ground state entropy; Potts antiferromagnets

Indexed keywords

COMPUTATIONAL METHODS; ENTROPY; FUNCTIONS; GRAPH THEORY; GROUND STATE; STATISTICAL MECHANICS;

GROUND-STATE DEGENERACY; GROUND-STATE ENTROPY; HOMEOMORPHIC STRUCTURES; POTTS ANTIFERROMAGNETS; STRIP GRAPHS; ZERO-TEMPERATURE PARTITION FUNCTION;

ANTIFERROMAGNETIC MATERIALS;

EID: 0032183370     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0378-4371(98)00359-8     Document Type: Article

References (38)

1. R.B. Potts, Proc. Camb. Phil. Soc. 48 (1952) 106.
2. F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235; errata, ibid. 55 (1983) 315.
3. We recall the strict mathematical definition of a graph as a finite collection of vertices and bonds connecting (all or a subset of the) vertices, such that there are (i) no loops, i.e., bonds connecting a vertex to itself and (ii) no multiple bonds connecting a given pair of vertices.
4. G.D. Birkhoff, Ann. Math. 14 (1912) 42; H. Whitney, Ann. Math. 33 (1932) 688; Bull. Am. Math. Soc. 38 (1932) 572; G.D. Birkhoff, D.C. Lewis, Trans. Am. Math. Soc. 60 (1946) 355.
5. G.D. Birkhoff, Ann. Math. 14 (1912) 42; H. Whitney, Ann. Math. 33 (1932) 688; Bull. Am. Math. Soc. 38 (1932) 572; G.D. Birkhoff, D.C. Lewis, Trans. Am. Math. Soc. 60 (1946) 355.
6. G.D. Birkhoff, Ann. Math. 14 (1912) 42; H. Whitney, Ann. Math. 33 (1932) 688; Bull. Am. Math. Soc. 38 (1932) 572; G.D. Birkhoff, D.C. Lewis, Trans. Am. Math. Soc. 60 (1946) 355.
7. G.D. Birkhoff, Ann. Math. 14 (1912) 42; H. Whitney, Ann. Math. 33 (1932) 688; Bull. Am. Math. Soc. 38 (1932) 572; G.D. Birkhoff, D.C. Lewis, Trans. Am. Math. Soc. 60 (1946) 355.
8. R.C. Read, J. Combin. Theory 4 (1968) 52; R.C. Read, W.T. Tutte, Chromatic Polynomials, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory, vol. 3, Academic Press, New York, 1988.
9. R.C. Read, J. Combin. Theory 4 (1968) 52; R.C. Read, W.T. Tutte, Chromatic Polynomials, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory, vol. 3, Academic Press, New York, 1988.
10. W.T. Tutte, in: G.C. Rota (Ed.), Graph Theory, Encyclopedia of Mathematics and its Applications, vol. 21, Addison-Wesley, New York, 1984.
11. F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969; N.L. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, U.K., 1st ed. 1974, 2nd ed. 1993.
12. F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969; N.L. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, U.K., 1st ed. 1974, 2nd ed. 1993.
13. L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, 1960, p. 466.
14. E.H. Lieb, F.Y. Wu, in: C. Domb, M.S. Green, (Eds.), Phase Transitions and Critical Phenomena, vol. 1, Academic Press, New York, 1972, p. 331.
15. Some related earlier work is in Refs. [11-14].
16. N.L. Biggs, R.M. Damerell, D.A. Sands, J. Combin. Theory B 12 (1972) 123; N.L. Biggs, G.H.J. Meredith, J. Combin. Theory B 20 (1976) 5.
17. N.L. Biggs, R.M. Damerell, D.A. Sands, J. Combin. Theory B 12 (1972) 123; N.L. Biggs, G.H.J. Meredith, J. Combin. Theory B 20 (1976) 5.
18. S. Beraha, J. Kahane, N.J. Weiss, J. Combin. Theory B 28 (1980) 52.
19. R.J. Baxter, J. Phys. A 20 (1987) 5241.
20. R.C. Read, G.F. Royle, in: Graph Theory, Combinatorics, and Applications, vol. 2, Wiley, New York, 1991, p. 1009.
21. R. Shrock, S.-H. Tsai, J. Phys. A 30 (1997) 495.
22. R. Shrock, S.-H. Tsai, Phys. Rev. E 55 (1997) 5165.
23. R. Shrock, S.-H. Tsai, Phys. Rev. E 56 (1997) 1342.
24. R. Shrock, S.-H. Tsai, Phys. Rev. E 56 (1997) 3935.
25. R. Shrock, S.-H. Tsai, Phys. Rev. E 55 (1997) 6791; E 56 (1997) 2733, 4111; S.-H. Tsai, Phys. Rev. E 57 (1998) 2686.
26. R. Shrock, S.-H. Tsai, Phys. Rev. E 55 (1997) 6791; E 56 (1997) 2733, 4111; S.-H. Tsai, Phys. Rev. E 57 (1998) 2686.
27. R. Shrock, S.-H. Tsai, Phys. Rev. E 55 (1997) 6791; E 56 (1997) 2733, 4111; S.-H. Tsai, Phys. Rev. E 57 (1998) 2686.
28. M. Roček, R. Shrock, S.-H. Tsai, Physica A 252 (1998) 505.
29. R. Shrock, S.-H. Tsai, Stony Brook preprints ITP-SB-98-17,18; see also Ref. [28].
30. j(q) for the coefficient functions in script N sign(x) in Eq. (2.5).
31. contr., q). The reverse process of bond deletion, leading to the same equation, is known as the deletion-contraction theorem.
32. y = 3 square strip).
33. R.C. Read, E.G. Whitehead, Chromatic polynomials of homeomorphism classes of graphs, University of Waterloo preprint; Discrete Math., to appear. We thank Professor Read for kindly giving us a copy of this preprint.
34. + such that P(G, q) > G.
35. S.-H. Tsai, Ph.D. thesis, State University of New York at Stony Brook, April, 1998.
36. y remaining finite.
37. In the technical terminology of algebraic geometry, a multiple point on an algebraic curve is a point where several branches of the curve intersect. See, e.g., R. Hartshorne, Algebraic Geometry, Springer, New York, 1977.
38. px).