Skip to main content
SpringerLink
Log in

Harmonic Functions on Metric Graphs Under the Anti-Kirchhoff Law

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

When does an infinite metric graph allow nonconstant bounded harmonic functions under the anti-Kirchhoff transition law? We give a complete answer to this question in the cases where Liouville’s theorem holds, for trees, for graphs with finitely many essential ramification nodes and for generalized lattices. It turns out that the occurrence of nonconstant bounded harmonic functions under the anti-Kirchhoff law differs strongly from the one under the classical continuity condition combined with the Kirchhoff incident flow law.

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. von Below, J., Mugnolo, D.: The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions. Linear Algebra Appl. 439, 1792–1814 (2013)

    Article  MathSciNet  Google Scholar 

  2. Mugnolo, D.: Vector-valued heat equations and networks with coupled dynamic boundary conditions. Adv. Differ. Equ. 15, 1125–1160 (2010)

    MathSciNet  MATH  Google Scholar 

  3. von Below, J., Lubary, J.A.: Harmonic functions on locally finite networks. Results Math. 45, 1–20 (2004)

    Article  MathSciNet  Google Scholar 

  4. Biggs, N.L.: Algebraic Graph Theory. Cambridge Tracts Mathematics, vol. 67, 1st edn. Cambridge University Press (1967)

  5. Biggs, N.L.: Algebraic Graph Theory. Cambridge Tracts Mathematics, vol. 67, 2nd edn. Cambridge University Press (1993)

  6. Mohar, B., Woess, W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21, 209–234 (1989)

    Article  MathSciNet  Google Scholar 

  7. Wilson, R.J.: Introduction to Graph Theory. Oliver & Boyd, Edinburgh (1972)

    MATH  Google Scholar 

  8. von Below, J., Lubary, J.A.: The eigenvalues of the Laplacian on locally finite networks. Results Math. 47, 199–225 (2005)

    Article  MathSciNet  Google Scholar 

  9. von Below, J.: A characteristic equation associated with an eigenvalue problem on \({C}^2\)-networks. Linear Algebra Appl. 71, 309–325 (1985)

    Article  MathSciNet  Google Scholar 

  10. von Below, J., Lubary, J.A., Vasseur, B.: Some remarks on the eigenvalue multiplicities of the Laplacian on infinite locally finite trees. Results Math. 63, 1331–1350 (2013)

    Article  MathSciNet  Google Scholar 

  11. von Below, J.: The index of a periodic graph. Results Math. 25, 198–223 (1994)

    Article  MathSciNet  Google Scholar 

  12. Collatz, L.: Spektren periodischer Graphen. Result. Math. 1, 42–53 (1979)

    Article  MathSciNet  Google Scholar 

  13. Woess, W.: Random Walks on Infinite Graphs and Groups, vol. 138. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

Download references

Acknowledgements

Joachim von Below is grateful to the research group GREDPA at UPC Barcelona for the invitation in 2018. José A. Lubary is grateful to the LMPA Joseph Liouville at ULCO in Calais for the invitation in 2018. The authors are indebted to the anonymous referee for valuable remarks.

Author information

Authors and Affiliations

  1. LMPA Joseph Liouville ULCO, FR CNRS Math. 2956, Universités Lille Nord de France, 50, rue F. Buisson, CS 80699, 62228, Calais Cedex, France

    Joachim von Below

  2. Departament de Matemàtiques, Universitat Politècnica de Catalunya, Campus Nord, Edifici A3, Jordi Girona, 1-3, 08034, Barcelona, Spain

    José A. Lubary

Authors
  1. Joachim von Below
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José A. Lubary
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joachim von Below.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

José A. Lubary was supported by MINECO Grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P and part of the Catalan research group 2017 SGR 1392.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

von Below, J., Lubary, J.A. Harmonic Functions on Metric Graphs Under the Anti-Kirchhoff Law. Results Math 74, 36 (2019). https://doi.org/10.1007/s00025-019-0966-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-019-0966-2

Keywords

Mathematics Subject Classification

Search

Navigation