Skip to main content
Log in

Cheeger constants, growth and spectrum of locally tessellating planar graphs

Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and global geometric invariants, namely the Cheeger constants and the exponential growth. We also discuss spectral applications.

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.

Institutional subscriptions

References

  1. Bartholdi L., Ceccherini-Silberstein T.G.: Salem numbers and growth series of some hyperbolic graphs. Geom. Dedicata 90, 107–114 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baues O., Peyerimhoff N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25(1), 141–159 (2001)

    MATH  MathSciNet  Google Scholar 

  3. Baues O., Peyerimhoff N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243–263 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cannon J.W., Wagreich P.: Growth functions on surface groups. Math. Ann. 293(2), 239–257 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. DeVos M., Mohar B.: An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture. Trans. Am. Math. Soc. 359(7), 3287–3300 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dodziuk J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Durrett, R., Pinsky, M.A. (eds.) Geometry of Random Motion, vol. 73, pp. 25–40. AMS Contemporary Mathematics (1988)

  8. Dodziuk J., Kendall W.S.: Combinatorial Laplacians and isoperimetric inequality. In: Elworthy, K.D. (eds) From Local Times to Global Geometry, Control and Physics, pp. 68–75. Longman Scientific and Technical, Harlow (1986)

    Google Scholar 

  9. Donnelly H., Li P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46(3), 497–503 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. Floyd W.J., Plotnick S.P.: Growth functions on Fuchsian groups and the Euler characteristic. Invent. Math. 88(1), 1–29 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fujiwara K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku Math. J. 48(2), 293–302 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fujiwara K.: Laplacians on rapidly branching trees. Duke Math. J. 83(1), 191–202 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, 2nd edn, Universitext. Springer-Verlag, Berlin (1990)

  14. Gromov, M.: Hyperbolic Groups. Essays in Group Theory, pp. 75–263. Math. Sci. Res. Inst. Publ. 8. Springer, New York (1987)

  15. Häggström O., Jonasson J., Lyons R.: Explicit isoperimetric constants and phase transitions in the random-cluster model. Ann. Probab. 30(1), 443–473 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Higuchi Y.: Combinatorial curvature for planar graphs. J. Graph Theory 38(4), 220–229 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Higuchi Y.: Boundary area growth and the spectrum of discrete Laplacian. Ann. Glob. Anal. Geom. 24(3), 201–230 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Higuchi Y., Shirai T.: Isoperimetric constants of (d,f)-regular planar graphs. Interdiscip. Inform. Sci. 9(2), 221–228 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Higuchi, Y., Shirai, T.: Some spectral and geometric properties for infinite graphs. In: Discrete Geometric Analysis, pp. 29–56. Contemp. Math. 347. Amer. Math. Soc., Providence (2004)

  20. Hoory S., Linial N., Wigderson A.: Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.) 43(4), 439–561 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Keller M.: The essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), 51–66 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. Preprint

  23. Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. In: Mathematical Modelling of Natural Phenomena, Spectral Problems (to appear)

  24. Klassert S., Lenz D., Peyerimhoff N., Stollmann P.: Elliptic operators on planar graphs: unique continuation for eigenfunctions and nonpositive curvature. Proc. Am. Math. Soc. 134, 1549–1559 (2005)

    Article  MathSciNet  Google Scholar 

  25. McKean H.P.: An upper bound to the spectrum of Δ on a manifold of negative curvature. J. Differ. Geom. 4, 359–366 (1970)

    MATH  MathSciNet  Google Scholar 

  26. Mohar B.: Light structures in infinite planar graphs without the strong isoperimetric property. Trans. Am. Math. Soc. 354(8), 3059–3074 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mohar B., Woess W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21(3), 209–234 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  28. Stone D.A.: A combinatorial analogue of a theorem of Myers and Correction to my paper: “A combinatorial analogue of a theorem of Myers”. Illinois J. Math. 20(1), 12–21 (1976) 551–554 (1976)

    MATH  MathSciNet  Google Scholar 

  29. Urakawa H.: The spectrum of an infinite graph. Can. J. Math. 52(5), 1057–1084 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph. http://arxiv.org/abs/0801.0812

  31. Woess W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124(3), 385–393 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

  33. Wojciechowski, R.K.: Stochastic completeness of graphs. PhD thesis http://arxiv.org/abs/0712.1570. (2007)

  34. Wojciechowski R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58(3), 1419–1441 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matthias Keller.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Keller, M., Peyerimhoff, N. Cheeger constants, growth and spectrum of locally tessellating planar graphs. Math. Z. 268, 871–886 (2011). https://doi.org/10.1007/s00209-010-0699-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-010-0699-0

Keywords

Navigation