Skip to main content
SpringerLink
Log in

A recursive method for calculating the total number of spanning trees and its applications in self-similar small-world scale-free network models

  • Regular Article
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

The problem of determining and calculating the number of spanning trees of any finite graph (model) is a great challenge, and has been studied in various fields, such as discrete applied mathematics, theoretical computer science, physics, chemistry and the like. In this paper, firstly, thank to lots of real-life systems and artificial networks built by all kinds of functions and combinations among some simpler and smaller elements (components), we discuss some helpful network-operation, including link-operation and merge-operation, to design more realistic and complicated network models. Secondly, we present a method for computing the total number of spanning trees. As an accessible example, we apply this method to space of trees and cycles respectively, and our results suggest that it is indeed a better one for such models. In order to reflect more widely practical applications and potentially theoretical significance, we study the enumerating method in some existing scale-free network models. On the other hand, we set up a class of new models displaying scale-free feature, that is to say, following P(k) ~ k−γ, where γ is the degree exponent. Based on detailed calculation, the degree exponent γ of our deterministic scale-free models satisfies γ > 3. In the rest of our discussions, we not only calculate analytically the solutions of average path length, which indicates our models have small-world property being prevailing in amounts of complex systems, but also derive the number of spanning trees by means of the recursive method described in this paper, which clarifies our method is convenient to research these models.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B.A. Huberman, The laws of the web (MIT Press, Cambridge, MA, USA, 2001)

  2. D. Garlaschelli, G. Caldarelli, L. Pietronero, Nature 423, 165 (2003)

    Article  ADS  Google Scholar 

  3. M.E. Newman, Phys. Rev. E 64, 016131 (2001)

    Article  ADS  Google Scholar 

  4. J.M. Montoya, R.V. Sole, J. Theor. Biol. 214, 405 (2002)

    Article  Google Scholar 

  5. A. Wagner, Mol. Biol. Evol. 18, 1283 (2001)

    Article  Google Scholar 

  6. H. Jong, B. Tombor, R. Albert, Z.N. Oltvai, A.-L. Barabási, Nature 407, 651 (2000)

    Article  ADS  Google Scholar 

  7. C.M. Song, T. Koren, P. Wang, A.L. Barabási, Nat. Phys. 1760, 1 (2010)

    Google Scholar 

  8. F. Ma, J. Su, B. Yao, M. Yao, in Mechanical and Electronic Engineering Conference (JIMEC 2016) (2016), Vol. 59, pp. 155–162

  9. A.-L. Barabási, E. Ravasz, T. Vicsek, Physica A 299, 559 (2001)

    Article  ADS  Google Scholar 

  10. E. Agliari, R. Burioni, Phys. Rev. E 80, 031125 (2009)

    Article  ADS  Google Scholar 

  11. F. Chung, L. Lu, Proc. Natl. Acad. Sci. USA 99, 15879 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  12. P. Erdös, A. Rényi, Publ. Math. 6, 290 2016

    Google Scholar 

  13. D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)

    Article  ADS  Google Scholar 

  14. D.J. Watts, Small worlds: the dynamics of networks between order and randomness (Princeton University Press, Princeton, NJ, 1999)

  15. A.-L. Barabási, R. Albert, H. Jeong, Physica A 272, 173 (1999)

    Article  ADS  Google Scholar 

  16. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Phys. Rev. 65, 066122 (2002)

    Google Scholar 

  17. B. Yao, F. Ma, J. Su, X.M. Wang, X.Y. Zhao, M. Yao, Proceedings of 2016 IEEE Advanced Information Management, Communicates, Electronic and Automation Control Conference (IMCEC2016) (2016), pp. 549–554

  18. F. Ma, B. Yao, Physica A 484, 182 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  19. Q.H. Chen, D.H. Shi, Physica A 335, 240 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  20. H.R. Liu, R.R. Yin, B. Liu, Y.Q. Li, Comput. Electr. Eng. 56,533 (2016)

    Article  Google Scholar 

  21. J.A. Bondy, U.S.R. Murty, Graph theory with application (MacMillan Press Ltd, London, Basingstoke, New York, 1976)

  22. Z.Z. Zhang, B. Wu, F. Comellas, Discret. Appl. Math. 169, 206 (2014)

    Article  Google Scholar 

  23. Z.Z. Zhang, H. Liu, B. Wu, S. Zhou, Europhys. Lett. 90, 1632 (2012)

    Google Scholar 

  24. Z.Z. Zhang, H. Liu, B. Wu, T. Zhou, Phys. Rev. 83, 016116 (2011)

    ADS  Google Scholar 

  25. F.Y. Wu, Physica A 10, 113 (1977)

    Google Scholar 

  26. Z.Z. Zhang, B. Wu, Y. Lin, Physica A 391, 3342 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  27. J. Wu, Y.J. Tan, H.Z. Deng, D.D. Zhu, Syst. Eng. Theory Pract. 27, 101 (2007)

    Article  Google Scholar 

  28. C.J. Colbourn, The combinatorics of network reliability (Oxford University Press, New York, 1987)

  29. F. Ma, B. Yao, Physica A 492, 2205 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  30. F. Ma, J. Su, Y.X. Hao, B. Yao, G.H. Yan, Physica A 492, 1194 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  31. G.J. Szabó, M. Alava, J. Kertész, Phys A. 330, 31 (2003)

    Article  MathSciNet  Google Scholar 

  32. N. Takashi, A.E. Motter, Phys. Rev. E. 73, 77 (2006)

    Google Scholar 

  33. P. Marchal, Electron. Commun. Probab. 5, 39 (2000)

    Article  Google Scholar 

  34. J.D. Noh, H. Rieger, Phys. Rev. Lett. 92, 118701 (2004)

    Article  ADS  Google Scholar 

  35. Z. Zhang, T. Shan, G. Chen, Phys. Rev. E 87, 012112 (2013)

    Article  ADS  Google Scholar 

  36. C. Godsil, G. Royle, Algebraic graph theory, in Graduate texts in mathematics (Springer, New York, 2001), Vol. 207

  37. J. Huang, S. Liu, Bull. Aust. Math. Soc. 91, 352 (2015)

    Article  Google Scholar 

  38. T. Nishikawa, A.E. Motter, Y.-C. Lai, F.C. Hoppensteadt, Phys. Rev. Lett. 91, 014101 (2003)

    Article  ADS  Google Scholar 

  39. S. Jung, S. Kim, B. Kahng, Phys. Rev. E 65, 056101 (2002)

    Article  ADS  Google Scholar 

  40. R. Cohen, S. Havlin, Phys. Rev. Lett. 90, 058701 (2003)

    Article  ADS  Google Scholar 

  41. R. Shrock, F.Y. Wu, Physica A 33, 3881 (2000)

    Google Scholar 

  42. Y. Xiao, H.X. Zhao, Physica A 392, 4576 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  43. Y. Xiao, H.X. Zhao, G.N. Hu, X.J. Ma, Physica 406, 236 (2014)

    Article  MathSciNet  Google Scholar 

  44. R. Lyons, R. Peled, O. Schramm, Combin. Probab. 17, 711 (2007)

    Google Scholar 

  45. S.C. Chang, L.C. Chen, J. Stat. Phys. 126, 649 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  46. Y. Lin, B. Wu, Z.Z. Zhang, G.R. Chen, J. Math. Phys. 52, 113303 (2011)

    Article  ADS  Google Scholar 

  47. K.I. Goh, G. Salvi, B. Kahng, Phys. Rev. Lett. 96, 018701 (2006)

    Article  ADS  Google Scholar 

  48. E. Teufl, S. Wagner, J. Phys. A 43, 415001 (2010)

    Article  MathSciNet  Google Scholar 

  49. F. Comellas, A. Miralles, H.X. Liu, Z.Z. Zhang, Physica A 392, 2803 (2013)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Electronics Engineering and Computer Science, Peking University, Beijing, 100871, P.R. China

    Fei Ma

  2. School of Software and Microelectronics, Peking University, Beijing, 102600, P.R. China

    Fei Ma

  3. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu, 730070, P.R. China

    Fei Ma, Jing Su & Bing Yao

  4. School of Electronics and Information Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, P.R. China

    Bing Yao

Authors
  1. Fei Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jing Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bing Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Ma.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ma, F., Su, J. & Yao, B. A recursive method for calculating the total number of spanning trees and its applications in self-similar small-world scale-free network models. Eur. Phys. J. B 91, 82 (2018). https://doi.org/10.1140/epjb/e2018-80560-8

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2018-80560-8

Keywords

Search

Navigation