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Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

  • Shu-Chiuan ChangEmail author
  • Robert Shrock
Article
  • 9 Downloads

Abstract

We infer upper and lower bounds on the exponential growth constants \(\alpha (\Lambda )\), \(\alpha _0(\Lambda )\), and \(\beta (\Lambda )\) describing the large-n behavior of, respectively, the number of acyclic orientations, acyclic orientations with a unique source vertex, and totally cyclic orientations of arrows on bonds of several n-vertex heteropolygonal Archimedean lattices \(\Lambda \). These are, to our knowledge, the best bounds on these growth constants. The inferred upper and lower bounds on the growth constants are quite close to each other, which enables us to infer rather accurate estimates for the actual exponential growth constants. Our new results for heteropolygonal Archimedean lattices, combined with our recent results for homopolygonal Archimedean lattices, are consistent with the inference that the exponential growth constants \(\alpha (\Lambda )\), \(\alpha _0(\Lambda )\), and \(\beta (\Lambda )\) on these lattices are monotonically increasing functions of the lattice coordination number. Comparisons are made with the corresponding growth constants for spanning trees on these lattices. Our findings provide further support for the Merino–Welsh and Conde–Merino conjectures.

Keywords

Acyclic orientations Cyclic orientations Directed graphs 

Notes

Acknowledgements

This research was supported in part by the Taiwan Ministry of Science and Technology grant MOST 103-2918-I-006-016 (S.-C.C.) and by the U.S. National Science Foundation grant No. NSF-PHY-16-1620628 (R.S.).

References

  1. 1.
    Chang, S.-C., Shrock, R.: Asymptotic behavior of acyclic and totally cyclic orientations of families of directed lattice graphs, arXiv:1810.07357
  2. 2.
    Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge, UK (1993)zbMATHGoogle Scholar
  3. 3.
    Welsh, D.J.A.: Complexity: Knots, Colourings, and Counting. Cambridge University Press, Cambridge (1993)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bollobás, B.: Modern Graph Theory. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    Chartrand, G., Lesniak, L.: Graphs and Digraphs. Chapman and Hall/CRC, New York (2005)zbMATHGoogle Scholar
  6. 6.
    Grünbaum, B., Shephard, G.C.: Tilings and Patterns: An Introduction. Freeman, New York (1989)zbMATHGoogle Scholar
  7. 7.
    For reviews of chromatic polynomials, see, e.g., R. C. Read and W. T. Tutte, “Chromatic Polynomials”, in Selected Topics in Graph Theory, 3, eds. L. W. Beineke and R. J. Wilson (Academic Press, New York, 1988), pp. 15-42 and F. M. Dong, K. M. Koh, and K. L. Teo, Chromatic Polynomials and Chromaticity of Graphs (World Scientific, Singapore, 2005)Google Scholar
  8. 8.
    Stanley, R.P.: Acyclic orientations of graphs. Discrete Math. 5, 171–178 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Greene, C., Zaslavsky, T.: On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs. Trans. Am. Math. Soc. 280, 97–126 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Tutte, W.T.: On dichromatic polynomials. J. Comb. Theory 2, 301–320 (1967)CrossRefzbMATHGoogle Scholar
  12. 12.
    Brylawski, T., Oxley, J.: The Tutte polynomial and its applications. In: White, N. (ed.) Matroid Applications. Encyclopedia of Mathematics and its Applications, vol. 40, pp. 123–225. Cambridge University Press, Cambridge (1992)Google Scholar
  13. 13.
    Welsh, D.J.A., Merino, C.: The Potts model and the Tutte polynomial. J. Math. Phys. 41, 1127–1152 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gebhard, D.D., Sagan, B.E.: Sinks in acyclic orientations of graphs. J. Comb. Theory B 80, 130–146 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Merino, C., Welsh, D.J.A.: Forest, colorings, and acyclic orientations of the square lattice. Ann. Comb. 3, 417–429 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Calkin, N., Merino, C., Noble, S., Noy, M.: Improved bounds for the number of forests and acyclic orientations in the square lattice. Electron. J. Comb. 10(R4), 1–18 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chang, S.-C., Shrock, R.: Tutte polynomials and related asymptotic limiting functions for recursive families of graphs (talk given by R. Shrock at Workshop on Tutte polynomials, Centre de Recerca Matemática (CRM), Sept. 2001, Univ. Autonoma de Barcelona), Adv. Appl. Math. 32, 44-87 (2004)Google Scholar
  18. 18.
    Las Vergnas, M.: Acyclic and totally cyclic orientations of combinatorial geometries. Discrete Math. 20, 51–61 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Las Vergnas, M.: Convexity in oriented matroids. J. Comb. Theory B 29, 231–243 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Biggs, N.L., Damerell, R.M., Sands, D.A.: Recursive families of graphs. J. Comb. Theory B 12, 123–131 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Beraha, S., Kahane, J., Weiss, N.: Limits of chromatic zeros of some families of maps. J. Comb. Theory B 28, 52–65 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Roček, M., Shrock, R., Tsai, S.-H.: Chromatic polynomials for families of strip graphs and their asymptotic limits. Phys. A 252, 505–546 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Shrock, R., Tsai, S.-H.: Ground state degeneracy of Potts antiferromagnets on 2D lattices: approach using infinite cyclic strip graphs. Phys. Rev. E 60, 3512–3515 (1999)ADSCrossRefGoogle Scholar
  24. 24.
    Shrock, R., Tsai, S.-H.: Exact partition functions for Potts antiferromagnets on cyclic lattice strips. Phys. A 275, 429–449 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Shrock, R.: Exact Potts model partition functions on strip graphs. Phys. A 283, 388–446 (2000)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Shrock, R., Tsai, S.-H.: Lower bounds and series for the ground state entropy of the Potts antiferromagnet on Archimedean lattices and their duals. Phys. Rev. E 56, 4111–4124 (1997)ADSCrossRefGoogle Scholar
  27. 27.
    Chang, S.-C., Wang, W.: Spanning trees on lattices and integral identities. J. Phys. A 39, 10263–10275 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Shrock, R.: Chromatic polynomials and their zeros and asymptotic limits for families of graphs. Discrete Math. 231, 421–446 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Shrock, R., Tsai, S.-H.: Asymptotic limits and zeros of chromatic polynomials and ground state entropy of Potts antiferromagnets. Phys. Rev. E 55, 5165–5179 (1997)ADSCrossRefGoogle Scholar
  30. 30.
    Shrock, R., Tsai, S.-H.: Upper and lower bounds for the ground state entropy of antiferromagnetic Potts models. Phys. Rev. E 55, 6791–6794 (1997)ADSCrossRefGoogle Scholar
  31. 31.
    Shrock, R., Tsai, S.-H.: Ground state entropy of antiferromagnetic Potts models: bounds, series, and Monte Carlo measurements. Phys. Rev. E 56, 2733–2737 (1997)ADSCrossRefGoogle Scholar
  32. 32.
    Chang, S.-C., Shrock, R.: Improved lower bounds on ground state entropy of the antiferromagnetic Potts model. Phys. Rev. E 91, 052142 (2015)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Biggs, N.L.: Colouring square lattice graphs. Bull. Lond. Math. Soc. 9, 54–56 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wu, F.Y.: Number of spanning trees on a lattice. J. Phys. A 10, L113–L115 (1977)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Shrock, R., Wu, F.Y.: Spanning trees on graphs and lattices in \(d\) dimensions. J. Phys. A 33, 3881–3902 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Chang, S.-C., Shrock, R.: Some exact results for spanning trees on lattices. J. Phys. A 39, 5653–5658 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Baxter, R.J.: Chromatic polynomials of large triangular lattices. J. Phys. A 20, 5241–5261 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Thomassen, C.: Spanning trees and orientations of graphs. J. Comb. 1, 101–111 (2010)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Conde, R., Merino, C.: Comparing the number of acyclic and totally cyclic orientations with that of spanning trees of a graph. Int. J. Math. Comb. 2, 79–89 (2009)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Merino, C., Ibañez, M., Guadalupe Rodrígez, M.: Guadalupe Rodrígez, a note on some inequalities for the Tutte polynomial of a matroid. Electron. Notes Direcrete Math. 34, 603–607 (2009)CrossRefzbMATHGoogle Scholar
  41. 41.
    Chávez-Lomeli, L.E., Merino, C., Noble, S.D., Ramírez-Ibáñez, M.: Some inequalities for the Tutte polynomial. Eur. J. Comb. 32, 422–433 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Noble, S.D., Royle, G.F.: The Merino–Welsh conjecture holds for series-parallel graphs. Eur. J. Comb. 38, 24–35 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Knauer, K., Martínez-Sandoval, L., Luis Ramírez-Alfonsín, J.: A Tutte polynomial inequality for lattice path matroids, arXiv:1510.00600
  44. 44.
    Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54, 235–268 (1982)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Fortuin, C.M., Kasteleyn, P.W.: On the random cluster model. Physica 57, 536–564 (1972)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of PhysicsNational Cheng Kung UniversityTainanTaiwan
  2. 2.C. N. Yang Institute for Theoretical Physics and Department of Physics and AstronomyStony Brook UniversityStony BrookUSA

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