Letters in Mathematical Physics

, Volume 108, Issue 2, pp 377–390 | Cite as

Relating zeta functions of discrete and quantum graphs

  • Jonathan Harrison
  • Tracy WeyandEmail author


We write the spectral zeta function of the Laplace operator on an equilateral metric graph in terms of the spectral zeta function of the normalized Laplace operator on the corresponding discrete graph. To do this, we apply a relation between the spectrum of the Laplacian on a discrete graph and that of the Laplacian on an equilateral metric graph. As a by-product, we determine how the multiplicity of eigenvalues of the quantum graph, that are also in the spectrum of the graph with Dirichlet conditions at the vertices, depends on the graph geometry. Finally we apply the result to calculate the vacuum energy and spectral determinant of a complete bipartite graph and compare our results with those for a star graph, a graph in which all vertices are connected to a central vertex by a single edge.


Quantum graph Zeta function Spectral graph theory 

Mathematics Subject Classification

05C99 81Q10 81Q35 



The authors would like to thank Gregory Berkolaiko and the anonymous referees, whose suggestions substantially simplified the presentation of the main result. JH would like to thank the University of Warwick for their hospitality during his sabbatical where some of the work was carried out. JH was supported by the Baylor University research leave program. This work was partially supported by a grant from the Simons Foundation (354583 to Jonathan Harrison).


  1. 1.
    Asratian, A.S., Denley, T.M.J., Häggkvist, R.: Bipartite Graphs and their Applications. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    von Below, J.: A characteristic equation associated to an eigenvalue problem on \(c^2\)-networks. Linear Algebra Appl. 71, 309–325 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berkolaiko, G., Harrison, J.M. and Wilson, J.H.: Mathematical aspects of vacuum energy on quantum graphs. J. Phys. A 42(2):025204, 20 (2009)Google Scholar
  4. 4.
    Berkolaiko, G. and Kuchment, P.: Introduction to quantum graphs, vol. 186 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2013)Google Scholar
  5. 5.
    Chung, F.: Spectral graph theory, vol. 92 of CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI (1997)Google Scholar
  6. 6.
    Friedli, F. and Karlsson, A.: Spectral zeta functions of graphs and the Riemann zeta function in the critical strip, preprint arXiv:1410.8010 (2016)
  7. 7.
    Fulling, S.A., Kaplan, L. and Wilson, J.H.: Vacuum energy and repulsive Casimir forces in quantum star graphs. Phys. Rev. A (3) 76(1):012118, 7 (2007)Google Scholar
  8. 8.
    Fulling, S.A., Kuchment, P. and Wilson, J.H.: Index theorems for quantum graphs. J. Phys. A 40(47):14165, 16 (2007)Google Scholar
  9. 9.
    Harrison, J.M. and Kirsten, K.: Zeta functions of quantum graphs. J. Phys. A 44(23):235301, 29 (2011)Google Scholar
  10. 10.
    Harrison, J.M., Kirsten, K. and Texier, C.: Spectral determinants and zeta functions of Schrödinger operators on metric graphs. J. Phys. A 45(12):125206, 14 (2012)Google Scholar
  11. 11.
    Harrison, J.M., Weyand, T. and Kirsten, K.: Zeta functions of the Dirac operator on quantum graphs. J. Math. Phys. 57:102301, 10 (2016)Google Scholar
  12. 12.
    Hashimoto, K.: Zeta functions of finite graphs and representation of p-adic groups. Adv. Stud. Pure Math. 15, 211–280 (1989)MathSciNetGoogle Scholar
  13. 13.
    Kuchment, P.: Quantum graphs I. Some basic structures. Waves Random Media 14(1), S107–S128 (2004)MathSciNetCrossRefzbMATHADSGoogle Scholar
  14. 14.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F. and Clark, C.W. (eds): NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC. Cambridge University Press, Cambridge (2010)Google Scholar
  15. 15.
    Pankrashkin, K.: Spectra of Schrödinger operators on equilateral quantum graphs. Lett. Math. Phys. 77(2), 139–154 (2006)MathSciNetCrossRefzbMATHADSGoogle Scholar
  16. 16.
    Stark, H.M., and Terras, A.A.: Zeta functions of finite graphs and coverings. Adv. Math. 121, 142–165 (1996)Google Scholar
  17. 17.
    Sunada, T.: \(L\)-functions in geometry and some applications. In: Shiohama, K., Sakai, T., Sunada, T. (eds.) Curvature and Topology of Riemannian Manifolds (Katata, 1985). Lecture Notes in Math, vol. 1201, pp. 266–284. Springer, Berlin (1986)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsBaylor UniversityWacoUSA
  2. 2.Department of MathematicsRose-Hulman Institute of TechnologyTerre HauteUSA

Personalised recommendations