Advertisement

Tutte Polynomials of Two Self-similar Network Models

  • Yunhua LiaoEmail author
  • Xiaoliang Xie
  • Yaoping Hou
  • M. A. Aziz-Alaoui
Article
  • 39 Downloads

Abstract

The Tutte polynomial T(Gxy) of a graph G, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in combinatorics and statistical physics. Graph operations have been extensively applied to model complex networks recently. In this paper, we study the Tutte polynomials of the diamond hierarchical lattices and a class of self-similar fractal models which can be constructed through graph operations. Firstly, we find out the behavior of the Tutte polynomial under k-inflation and k-subdivision which are two graph operations. Secondly, we compute and gain the Tutte polynomials of this two self-similar fractal models by using their structure characteristic. Moreover, as an application of the obtained results, some evaluations of their Tutte polynomials are derived, such as the number of spanning trees and the number of spanning forests.

Keywords

Tutte polynomial The number of spanning trees Complex network model Subdivision Inflation 

Notes

Acknowledgements

The authors thank the anonymous referees for their valuable comments and helpful suggestions, which have considerably improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (No. 11571101), the Scientific Research Fund of Hunan Provincial Education Department (No. 16C0872) and the Hunan Provincial Natural Science Foundation of China (No. 2018JJ3255). Yunhua Liao and M. A. Aziz-Alaoui were supported by Normandie region France and the XTerm ERDF project (European Regional Development Fund) on Complex Networks and Applications.

References

  1. 1.
    Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Potts, R.B.: Some generalized order-disorder transformations. Math. Proc. Camb. Philos. Soc. 48(1), 106–109 (1952)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54(1), 235–268 (1982)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of 1/f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)ADSCrossRefGoogle Scholar
  5. 5.
    Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64, 1613–1616 (1990)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Jin, X.A., Zhang, F.J.: Zeros of the Jones polynomial for multiple crossing-twisted links. J. Stat. Phys. 140(6), 1054–1064 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ellis-Monaghan, J.A., Merino, C.: Graph polynomials and their applications I: the Tutte polynomial. In: Dehmer, M. (ed.) Structural Analysis of Complex Networks, pp. 219–255. Birkhüser, Boston (2011)CrossRefGoogle Scholar
  8. 8.
    Welsh, D.J.A., Merino, C.: The Potts model and the Tutte polynomial. J. Math. Phys. 41, 1127–1152 (2000)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Griffiths, R.B., Kaufman, M.: First-order transitions in defect structures at a second-order critical point for the Potts model on hierarchical lattices. Phys. Rev. B 26, 5022–5032 (1982)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hu, B.: Problem of universality in phase transitions on hierarchical lattices. Phys. Rev. Lett. 55, 2316–2319 (1985)ADSCrossRefGoogle Scholar
  11. 11.
    Yang, Z.R.: Family of diamond-type hierarchical lattices. Phys. Rev. B 38, 728–731 (1988)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Qin, Y., Yang, Z.R.: Diamond-type hierarchical lattices for the Potts antiferromagnet. Phys. Rev. B 43, 8576–8582 (1991)ADSCrossRefGoogle Scholar
  13. 13.
    de Silva, L.: Criticality and multifractality of the Potts ferromagnetic model on fractal lattices. Phys. Rev. B 53, 6345–6354 (1996)ADSCrossRefGoogle Scholar
  14. 14.
    Muzy, P.T., Salinas, S.R.: Ferromagnetic Potts model on a hierarchical lattice with random layered interactions. Int. J. Mod. Phys. B 4, 397–409 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    Bleher, P.M., Lyubich, M.Y.: Julia Sets and complex singularities in hierarchical Ising models. Commun. Math. Phys. 141, 453–474 (1991)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Qiao, J.Y.: Julia sets and complex singularities in diamondlike hierarchical Potts models. Sci. China Ser. A 48, 388–412 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ma, F., Yao, B.: The number of spanning trees of self-similar fractal models. Inf. Process. Lett. 136, 64–69 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cvetković, D., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  19. 19.
    Chang, S.C., Shrock, R.: Structure of the partition function and transfer matrices for the Potts model in a magnetic field on lattice strips. J. Stat. Phys. 137(4), 667–699 (2009)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Shrock, R., Xu, Y.: The structure of chromatic polynomials of planar triangulation graphs and implications for chromatic zeros and asymptotic limiting quantities. J. Phys. A 45(21), 215202 (2012)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Chang, S.C., Shrock, R.: Exact partition functions for the q-state Potts model with a generalized magnetic field on lattice strip graphs. J. Stat. Phys. 161(4), 915–932 (2015)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Alvarez, P.D., Canfora, F., Reyes, S.A., Riquelme, S.: Potts model on recursive lattices: some new exact results. Eur. Phys. J. B 85(3), 99 (2012)ADSCrossRefGoogle Scholar
  23. 23.
    Peng, J.H., Xiong, J., Xu, G.A.: Tutte polynomial of pseudofractal scale-free web. J. Stat. Phys. 159(5), 1196–1215 (2015)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Liao, Y.H., Hou, Y.P., Shen, X.L.: Tutte polynomial of the apollonian network. J. Stat. Mech. 10, P10043 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Chen, H.L., Deng, H.Y.: Tutte polynomial of scale-free networks. J. Stat. Phys. 163(4), 714–732 (2016)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Gong, H.L., Jin, X.A.: A general method for computing Tutte polynomials of self-similar graphs. Physica A 483, 117–129 (2017)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Lin, Y., Wu, B., Zhang, Z.Z., Chen, G.: Counting spanning trees in self-similar networks by evaluating determinants. J. Math. Phys. 52, 113303 (2011)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Rozenfeld, H., Havlin, S., Ben-Avraham, D.: Fractal and transfractal recursive scale-free net. New J. Phys. 9, 175 (2007)ADSCrossRefGoogle Scholar
  29. 29.
    Jing, H., Shu, C.L.: On the normalized Laplacian, degree-Kirchhoff index and spanning trees of graphs. Bull. Aust. Math. Soc. 91, 353–367 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Yunhua Liao
    • 1
    • 3
    Email author
  • Xiaoliang Xie
    • 1
  • Yaoping Hou
    • 2
  • M. A. Aziz-Alaoui
    • 3
  1. 1.Department of MathematicsHunan University of CommerceChangshaChina
  2. 2.Department of MathematicsHunan Normal UniversityChangshaChina
  3. 3.Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, ISCNLe HavreFrance

Personalised recommendations