Skip to main content
SpringerLink
Log in

Asymptotic behavior of the number of Eulerian orientations of graphs

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

The class of simple graphs with large algebraic connectivity (the second minimal eigenvalue of the Laplacian matrix) is considered. For graphs of this class, the asymptotic behavior of the number of Eulerian orientations is obtained. New properties of the Laplacian matrix are established, as well as an estimate of the conditioning of matrices with asymptotic diagonal dominance is obtained.

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. D. McKay, “The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs,” Combinatorica 10(4), 367–377 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Mihail and P. Winkler, “On the number of Eulerian orientations of a graph,” Algorithmica 16(4–5), 402–414 (1996).

    MathSciNet  MATH  Google Scholar 

  3. M. Las Vergnas, “Le polynôme de Martin d’un graphe eulérien,” Ann. Discrete Math. 17, 397–411 (1983).

    MATH  Google Scholar 

  4. A. Schrijver, “Bounds on the number of Eulerian orientations,” Combinatorica 3(3–4), 375–380 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Las Vergnas, “An upper bound for the number of Eulerian orientations of a regular graph,” Combinatorica 10(1), 61–65 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Fiedler, “Algebraic connectivity of graphs,” Czechoslovak Math. J. 23(98), 298–305 (1973).

    MathSciNet  Google Scholar 

  7. B. Mohar, “The Laplacian spectrum of graphs,” in Graph Theory, Combinatorics, and Applications, Wiley-Intersci. Publ. (John Wiley & Sons, New York, 1991), Vol. 2, pp. 871–898.

    Google Scholar 

  8. G. Kirchhoff, “Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird,” Ann. Phys. Chem. 148(12), 497–508 (1847).

    Article  Google Scholar 

  9. B. D. McKay and R. W. Robinson, “Asymptotic enumeration Eulerian circuits in the complete graph,” Combin. Probab. Comput. 7(4), 437–449 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Isaev, “Asymptotic behaviour of the number of Eulerian circuits,” Electron. J. Combin. 18(219) (2011).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Moscow, Russia

    M. I. Isaev

  2. Centre de Mathématiques Appliquées, École Polytechnique, Palaiseau, France

    M. I. Isaev

Authors
  1. M. I. Isaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. I. Isaev.

Additional information

Original Russian Text © M. I. Isaev, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 6, pp. 828–843.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Isaev, M.I. Asymptotic behavior of the number of Eulerian orientations of graphs. Math Notes 93, 816–829 (2013). https://doi.org/10.1134/S0001434613050210

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434613050210

Keywords

Search

Navigation