Annals of Global Analysis and Geometry

, Volume 24, Issue 3, pp 201–230

Boundary Area Growth and the Spectrum of Discrete Laplacian

  • Yusuke Higuchi
Article

Abstract

We introduce the boundary area growth as a new quantity for an infinite graph. Using this, we give some upper bounds for the bottom of the spectrum of the discrete Laplacian which relates closely to the transition operator. We also give some applications and examples.

discrete spectral geometry discrete Laplacian growth function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bollobás, B.: Graph Theory - An Introductory Course, Springer-Verlag, Berlin, 1985.Google Scholar
  2. 2.
    Bondy, J. A. and Murty, U. S. R.: Graph Theory with Its Applications, Macmillan, New York, 1976.Google Scholar
  3. 3.
    Brooks, R.: A relation between growth and the spectrum of the Laplacian, Math. Z. 178 (1981), 501–508.Google Scholar
  4. 4.
    Dodziuk, J. and Karp, L.: Spectral and function theory for combinatorial Laplacians, Contemp. Math. b73 (1988), 25–40.Google Scholar
  5. 5.
    Dodziuk, J. and Kendall, W. S.: Combinatorial Laplacians and isoperimetric inequality, in: K. D. Elworthy (ed.), From Local Times to Global Geometry, Control and Physics, Pitman Res. Notes Math. Ser. 150, Longman, Harlow, 1986, pp. 68–74.Google Scholar
  6. 6.
    Fujiwara, K.: Growth and the spectrum of the Laplacian on an infinite graph, Tôhoku Math. J. 48 (1996), 293–302.Google Scholar
  7. 7.
    Fujiwara, K.: Laplacians on rapidly branching trees, Duke Math. J. 83 (1996), 192–202.Google Scholar
  8. 8.
    Higuchi, Yu.: A remark on exponential growth and the spectrum of the Laplacian, Kodai Math. J. 24 (2001), 42–47.Google Scholar
  9. 9.
    Kesten, H.: Symmetric random walks on groups, Trans. Amer. Math. Soc. 92 (1959), 336–354.Google Scholar
  10. 10.
    Lyons, T.: A simple criterion for transience of a reversible Markov chain, Ann. Probab. 11 (1983), 393–402.Google Scholar
  11. 11.
    Nash-Williams, C. St. J. A.: Random walks and electric currents in networks, Proc. Cambridge Philos. Soc. 55 (1959), 181–194.Google Scholar
  12. 12.
    Sunada, T.: private communication.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Yusuke Higuchi
    • 1
  1. 1.Mathematics Laboratories, College of Arts and SciencesShowa UniversityFujiyoshida, YamanashiJapan

Personalised recommendations