Mathematische Zeitschrift

, Volume 268, Issue 3–4, pp 871–886 | Cite as

Cheeger constants, growth and spectrum of locally tessellating planar graphs



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.


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  1. 1.
    Bartholdi L., Ceccherini-Silberstein T.G.: Salem numbers and growth series of some hyperbolic graphs. Geom. Dedicata 90, 107–114 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baues O., Peyerimhoff N.: Curvature and geometry of tessellating plane graphs. Discrete Comput. Geom. 25(1), 141–159 (2001)MATHMathSciNetGoogle Scholar
  3. 3.
    Baues O., Peyerimhoff N.: Geodesics in non-positively curved plane tessellations. Adv. Geom. 6(2), 243–263 (2006)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cannon J.W., Wagreich P.: Growth functions on surface groups. Math. Ann. 293(2), 239–257 (1992)MATHCrossRefMathSciNetGoogle Scholar
  5. 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)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dodziuk J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)MATHCrossRefMathSciNetGoogle Scholar
  7. 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)Google Scholar
  8. 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. 9.
    Donnelly H., Li P.: Pure point spectrum and negative curvature for noncompact manifolds. Duke Math. J. 46(3), 497–503 (1979)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Floyd W.J., Plotnick S.P.: Growth functions on Fuchsian groups and the Euler characteristic. Invent. Math. 88(1), 1–29 (1987)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fujiwara K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku Math. J. 48(2), 293–302 (1996)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fujiwara K.: Laplacians on rapidly branching trees. Duke Math. J. 83(1), 191–202 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, 2nd edn, Universitext. Springer-Verlag, Berlin (1990)Google Scholar
  14. 14.
    Gromov, M.: Hyperbolic Groups. Essays in Group Theory, pp. 75–263. Math. Sci. Res. Inst. Publ. 8. Springer, New York (1987)Google Scholar
  15. 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)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Higuchi Y.: Combinatorial curvature for planar graphs. J. Graph Theory 38(4), 220–229 (2001)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Higuchi Y.: Boundary area growth and the spectrum of discrete Laplacian. Ann. Glob. Anal. Geom. 24(3), 201–230 (2003)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Higuchi Y., Shirai T.: Isoperimetric constants of (d,f)-regular planar graphs. Interdiscip. Inform. Sci. 9(2), 221–228 (2003)MATHCrossRefMathSciNetGoogle Scholar
  19. 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)Google Scholar
  20. 20.
    Hoory S., Linial N., Wigderson A.: Expander graphs and their applications. Bull. Am. Math. Soc. (N.S.) 43(4), 439–561 (2006)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Keller M.: The essential spectrum of the Laplacian on rapidly branching tessellations. Math. Ann. 346(1), 51–66 (2010)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. PreprintGoogle Scholar
  23. 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)Google Scholar
  24. 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)CrossRefMathSciNetGoogle Scholar
  25. 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)MATHMathSciNetGoogle Scholar
  26. 26.
    Mohar B.: Light structures in infinite planar graphs without the strong isoperimetric property. Trans. Am. Math. Soc. 354(8), 3059–3074 (2002)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Mohar B., Woess W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21(3), 209–234 (1989)MATHCrossRefMathSciNetGoogle Scholar
  28. 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)MATHMathSciNetGoogle Scholar
  29. 29.
    Urakawa H.: The spectrum of an infinite graph. Can. J. Math. 52(5), 1057–1084 (2000)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Weber, A.: Analysis of the physical Laplacian and the heat flow on a locally finite graph.
  31. 31.
    Woess W.: A note on tilings and strong isoperimetric inequality. Math. Proc. Camb. Philos. Soc. 124(3), 385–393 (1998)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000)Google Scholar
  33. 33.
    Wojciechowski, R.K.: Stochastic completeness of graphs. PhD thesis (2007)
  34. 34.
    Wojciechowski R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58(3), 1419–1441 (2009)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Mathematical InstituteFriedrich-Schiller-University JenaJenaGermany
  2. 2.Department of Mathematical SciencesUniversity of DurhamDurhamUK

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