Skip to main content
Log in

A New Proof of a Formula for the Bartholdi Zeta Function of a Digraph

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

We give a proof of a formula for the Bartholdi zeta function of a digraph by the method of Stark and Terras (Adv. Math. 121:124–165, 1996).

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bartholdi, L.: Counting paths in graphs. Enseign. Math. 45, 83–131 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Bass, H.: The Ihara-Selberg zeta function of a tree lattice. Int. J. Math. 3, 717–797 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Foata, D., Zeilberger, D.: A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs. Trans. Am. Math. Soc. 351, 2257–2274 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Grigorchuk R.I.: Symmetric random walks on discrete groups, Adv. Probab. Rel. Top. (D. Griffeath ed.) vol. 6, M. Dekker 1980, 285-325, pp. 132–152

  5. Hashimoto, K.: Zeta functions of finite graphs and representations of p-adic groups. In: Advanced Studies in Pure Mathematics, vol. 15, pp. 211–280. Academic Press, New York (1989)

  6. Ihara, Y.: On discrete subgroups of the two by two projective linear group over \(p\)-adic fields. J. Math. Soc. Japan 18, 219–235 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7, 7–25 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Mizuno, H., Sato, I.: Bartholdi zeta functions of graph coverings. J. Comb. Theory Ser. B 89, 27–41 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mizuno, H., Sato, I.: Bartholdi zeta function of a digraph. Eur. J. Comb. 24, 947–954 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings. Adv. Math. 121, 124–165 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tarfulea, A., Perlis, R.: Generalizing Ihara’s formula to zeta functions of partially directed graphs. Linear Algebra Appl. 431, 73–85 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank the referee for many valuable comments and many helpful suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Iwao Sato.

Additional information

I. Sato: Supported by Grant-in-Aid for Science Research (C).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sato, I. A New Proof of a Formula for the Bartholdi Zeta Function of a Digraph. Graphs and Combinatorics 32, 1571–1583 (2016). https://doi.org/10.1007/s00373-015-1668-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-015-1668-5

Keywords

Mathematics Subject Classification

Navigation