Graphs and Combinatorics

, Volume 31, Issue 6, pp 2215–2230 | Cite as

A Balanced Signed Digraph

Original Paper

Abstract

We extend a balanced signed graph to a digraph, and present a necessary and sufficient condition for a signed digraph to be balanced. Moreover, we give another necessary and sufficient condition for a signed digraph \((D,w)\) to be balanced by using zeta functions of \(D\). As an application, we discuss the structure of balanced coverings of signed digraphs under consideration of coverings of strongly connected digraphs.

Keywords

Balanced signed digraph Strongly connectivity Covering  Zeta function 

References

  1. 1.
    Bass, H.: The Ihara-Selberg zeta function of a tree lattice. Int. J. Math. 3, 717–797 (1992)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bowen, R., Lanford, O.: Zeta functions of restrictions of the shift transformation. Proc. Symp. Pure Math. 14, 43–49 (1970)MathSciNetCrossRefMATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Deng, A., Wu, Y.: Characteristic polynomials of digraphs having a semi-free action. Linear Algebra Appl. 408, 189–206 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Deng, A., Sato, I., Wu, Y.: Characteristic polynomials of ramified uniform covering digraphs. Eur. J. Comb. 28, 1099–1114 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gross, J.L., Tucker, T.W.: Generating all graph coverings by permutation voltage assignments. Discret. Math. 18, 273–283 (1977)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gross, J.L., Tucker, T.W.: Topological Graph Theory. Wiley, New York (1987)MATHGoogle Scholar
  8. 8.
    Harary, F.: On the notion of balance of a signed graph. Mich. Math. J. 2, 143–146 (1955)MathSciNetMATHGoogle Scholar
  9. 9.
    Hashimoto, K.: Zeta functions of finite graphs and representations of \(p\)-adic groups. Adv. Stud. Pure Math. vol. 15, pp. 211–280. Academic Press, New York (1989)Google Scholar
  10. 10.
    Ihara, Y.: On discrete subgroups of the two by two projective linear group over \(p\)-adic fields. J. Math. Soc. Jpn. 18, 219–235 (1966)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7, 7–25 (2000)MathSciNetMATHGoogle Scholar
  12. 12.
    Mizuno, H., Sato, I.: Zeta functions of digraphs. Linear Algebra Appl. 336, 181–190 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mizuno, H., Sato, I.: Weighted zeta functions of digraphs. Linear Algebra Appl. 355, 35–48 (2002)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mizuno, H., Sato, I.: Weighted Bartholdi zeta functions of digraphs. Far East J. Math. Sci. 27, 323–344 (2007)MathSciNetMATHGoogle Scholar
  15. 15.
    Negami, S.: The spherical genus and virtually planar graphs. Discret. Math. 70, 159–168 (1988)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Sato, I.: Bartholdi zeta functions of group coverings of digraphs. Far East J. Math. Sci. 18, 321–339 (2005)MathSciNetMATHGoogle Scholar
  17. 17.
    Sato, I.: Coverings of digraphs. Far East J. Math. Sci. 23, 281–293 (2006)MathSciNetMATHGoogle Scholar
  18. 18.
    Sato, I.: Weighted zeta functions of graph coverings. Electron. J. Comb. 13, R 91 (2006)Google Scholar
  19. 19.
    Serre, J.-P.: Linear Representations of Finite Group. Springer, New York (1977)CrossRefGoogle Scholar
  20. 20.
    Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings. Adv. Math. 121, 124–165 (1996)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings, part III. Adv. Math. 208, 467–489 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sunada, T.: \(L\)-functions in geometry and some applications. Lecture Notes in Math., vol. 1201, pp. 266–284. Springer, New York (1986)Google Scholar
  23. 23.
    Sunada, T.: Fundamental Groups and Laplacians (in Japanese). Kinokuniya, Tokyo (1988)MATHGoogle Scholar
  24. 24.
    Tarfulea, A., Perlis, R.: An Ihara formula for partially directed graphs. Linear Algebra Appl. 431, 73–85 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Mathematics Laboratories, College of Arts and SciencesShowa UniversityFujiyoshidaJapan
  2. 2.Oyama National College of TechnologyOyamaJapan

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