Skip to main content
SpringerLink
Log in

Introductory Graph and Metric Notions

  • Chapter
  • First Online:
Coarse Geometry and Randomness

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 2100))

  • 1643 Accesses

Abstract

In this section we start by reviewing some geometric properties of graphs. Those will be related to the behavior of random processes on the graphs in later sections.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For every choice of v ∈ V and r > 0.

  2. 2.

    A graph that can be embedded in the plane.

  3. 3.

    The number of vertices at distance ≤ n from some fixed vertex growth polynomially.

References

  1. I. Benjamini, C. Hoffman, ω-periodic graphs. Electron. J. Combin. 12, Research Paper 46, 12 pp. (2005) (electronic)

    Google Scholar 

  2. B. Bollobás, I. Leader, An isoperimetric inequality on the discrete torus. SIAM J. Discrete Math. 3(1), 32–37 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. B.H. Bowditch, A short proof that a subquadratic isoperimetric inequality implies a linear one. Mich. Math. J. 42(1), 103–107 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bonk, O. Schramm, Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10(2), 266–306 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. I. Benjamini, O. Schramm, Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality, in Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1850 (Springer, Berlin, 2004), pp. 73–76

    Google Scholar 

  6. I. Benjamini, O. Schramm, Á. Timár, On the separation profile of infinite graphs. Groups Geom. Dyn. 6(4), 639–658 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Eskin, D. Fisher, K. Whyte, Quasi-isometries and rigidity of solvable groups. Pure Appl. Math. Q. 3(4, part 1), 927–947 (2007)

    Google Scholar 

  8. A. Erschler, On drift and entropy growth for random walks on groups. Ann. Probab. 31(3), 1193–1204 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. J.F.L. Gall, Uniqueness and universality of the Brownian map. Arxiv preprint arXiv:1105.4842 (2011)

    Google Scholar 

  10. M. Gromov, P. Pansu, M. Katz, S. Semmes, Metric Structures for Riemannian and non-Riemannian Spaces, vol. 152 (Birkhäuser, Boston, 2006)

    Google Scholar 

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

    Google Scholar 

  12. B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth. J. Am. Math. Soc. 23(3), 815–829 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Kozma, Percolation, perimetry, planarity. Rev. Mat. Iberoam. 23(2), 671–676 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. T.J. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality. Geom. Funct. Anal. 10(1), 111–123 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.R. Lee, Y. Peres, Harmonic maps on amenable groups and a diffusive lower bound for random walks. Arxiv preprint arXiv:0911.0274 (2009)

    Google Scholar 

  16. R.J. Lipton, R.E. Tarjan, A separator theorem for planar graphs. SIAM J. Appl. Math. 36(2), 177–189 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Meier, Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups. London Mathematical Society Student Texts, vol. 73 (Cambridge University Press, Cambridge, 2008)

    Google Scholar 

  18. G. Miermont, The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210(2), 319–401 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Muchnik, I. Pak, Percolation on Grigorchuk groups. Commun. Algebra 29(2), 661–671 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. G.L. Miller, S.-H. Teng, W. Thurston, S.A. Vavasis, Geometric separators for finite-element meshes. SIAM J. Sci. Comput. 19(2), 364–386 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. V. Nekrashevych, Self-Similar Groups. Mathematical Surveys and Monographs, vol. 117 (American Mathematical Society, Providence, 2005)

    Google Scholar 

  22. V. Nekrashevych, G. Pete, Scale-invariant groups. Groups Geom. Dyn. 5(1), 139–167 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Peled, On rough isometries of Poisson processes on the line. Ann. Appl. Probab. 20(2), 462–494 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. G. Pete, Probability and Geometry on Groups (2009) (preprint)

    Google Scholar 

  25. B. Virág, Anchored expansion and random walk. Geom. Funct. Anal. 10(6), 1588–1605 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. N. Th. Varopoulos, L. Saloff-Coste, T. Coulhon, Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100 (Cambridge University Press, Cambridge, 1992)

    Google Scholar 

  27. W. Woess, Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions. Comb. Probab. Comput. 14(3), 415–433 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Weizmann Institute of Science, Rehovot, Israel

    Itai Benjamini

Authors
  1. Itai Benjamini
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Benjamini, I. (2013). Introductory Graph and Metric Notions. In: Coarse Geometry and Randomness. Lecture Notes in Mathematics(), vol 2100. Springer, Cham. https://doi.org/10.1007/978-3-319-02576-6_1

Download citation

Publish with us

Policies and ethics

Search

Navigation