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Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices

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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.

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References

  1. Chang, S.-C., Shrock, R.: Asymptotic behavior of acyclic and totally cyclic orientations of families of directed lattice graphs, arXiv:1810.07357

  2. Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge, UK (1993)

    Google Scholar 

  3. Welsh, D.J.A.: Complexity: Knots, Colourings, and Counting. Cambridge University Press, Cambridge (1993)

    Book  Google Scholar 

  4. Bollobás, B.: Modern Graph Theory. Springer, New York (1998)

    Book  Google Scholar 

  5. Chartrand, G., Lesniak, L.: Graphs and Digraphs. Chapman and Hall/CRC, New York (2005)

    Google Scholar 

  6. Grünbaum, B., Shephard, G.C.: Tilings and Patterns: An Introduction. Freeman, New York (1989)

    Google Scholar 

  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)

  8. Stanley, R.P.: Acyclic orientations of graphs. Discrete Math. 5, 171–178 (1973)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  10. Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)

    Article  MathSciNet  Google Scholar 

  11. Tutte, W.T.: On dichromatic polynomials. J. Comb. Theory 2, 301–320 (1967)

    Article  Google Scholar 

  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. Welsh, D.J.A., Merino, C.: The Potts model and the Tutte polynomial. J. Math. Phys. 41, 1127–1152 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  14. Gebhard, D.D., Sagan, B.E.: Sinks in acyclic orientations of graphs. J. Comb. Theory B 80, 130–146 (2000)

    Article  MathSciNet  Google Scholar 

  15. Merino, C., Welsh, D.J.A.: Forest, colorings, and acyclic orientations of the square lattice. Ann. Comb. 3, 417–429 (1999)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

  18. Las Vergnas, M.: Acyclic and totally cyclic orientations of combinatorial geometries. Discrete Math. 20, 51–61 (1977)

    Article  MathSciNet  Google Scholar 

  19. Las Vergnas, M.: Convexity in oriented matroids. J. Comb. Theory B 29, 231–243 (1980)

    Article  MathSciNet  Google Scholar 

  20. Biggs, N.L., Damerell, R.M., Sands, D.A.: Recursive families of graphs. J. Comb. Theory B 12, 123–131 (1972)

    Article  MathSciNet  Google Scholar 

  21. Beraha, S., Kahane, J., Weiss, N.: Limits of chromatic zeros of some families of maps. J. Comb. Theory B 28, 52–65 (1980)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  24. Shrock, R., Tsai, S.-H.: Exact partition functions for Potts antiferromagnets on cyclic lattice strips. Phys. A 275, 429–449 (2000)

    Article  MathSciNet  Google Scholar 

  25. Shrock, R.: Exact Potts model partition functions on strip graphs. Phys. A 283, 388–446 (2000)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  27. Chang, S.-C., Wang, W.: Spanning trees on lattices and integral identities. J. Phys. A 39, 10263–10275 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  28. Shrock, R.: Chromatic polynomials and their zeros and asymptotic limits for families of graphs. Discrete Math. 231, 421–446 (2001)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  32. Chang, S.-C., Shrock, R.: Improved lower bounds on ground state entropy of the antiferromagnetic Potts model. Phys. Rev. E 91, 052142 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  33. Biggs, N.L.: Colouring square lattice graphs. Bull. Lond. Math. Soc. 9, 54–56 (1977)

    Article  MathSciNet  Google Scholar 

  34. Wu, F.Y.: Number of spanning trees on a lattice. J. Phys. A 10, L113–L115 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  35. Shrock, R., Wu, F.Y.: Spanning trees on graphs and lattices in \(d\) dimensions. J. Phys. A 33, 3881–3902 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  36. Chang, S.-C., Shrock, R.: Some exact results for spanning trees on lattices. J. Phys. A 39, 5653–5658 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  37. Baxter, R.J.: Chromatic polynomials of large triangular lattices. J. Phys. A 20, 5241–5261 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  38. Thomassen, C.: Spanning trees and orientations of graphs. J. Comb. 1, 101–111 (2010)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  42. Noble, S.D., Royle, G.F.: The Merino–Welsh conjecture holds for series-parallel graphs. Eur. J. Comb. 38, 24–35 (2014)

    Article  MathSciNet  Google Scholar 

  43. Knauer, K., Martínez-Sandoval, L., Luis Ramírez-Alfonsín, J.: A Tutte polynomial inequality for lattice path matroids, arXiv:1510.00600

  44. Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54, 235–268 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  45. Fortuin, C.M., Kasteleyn, P.W.: On the random cluster model. Physica 57, 536–564 (1972)

    Article  ADS  MathSciNet  Google Scholar 

Download references

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.).

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Authors and Affiliations

  1. Department of Physics, National Cheng Kung University, Tainan, 70101, Taiwan

    Shu-Chiuan Chang

  2. C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY, 11794, USA

    Robert Shrock

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  1. Shu-Chiuan Chang
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  2. Robert Shrock
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Correspondence to Shu-Chiuan Chang.

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Communicated by Abhishek Dhar.

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Chang, SC., Shrock, R. Study of Exponential Growth Constants of Directed Heteropolygonal Archimedean Lattices. J Stat Phys 174, 1288–1315 (2019). https://doi.org/10.1007/s10955-019-02235-1

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