Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Lack of Ground State for NLSE on Bridge-Type Graphs Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Intrinsic Metrics on Graphs: A Survey

Author : Matthias Keller

Published in: Mathematical Technology of Networks

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

A few years ago various disparities for Laplacians on graphs and manifolds were discovered. The corresponding results are mostly related to volume growth in the context of unbounded geometry. Indeed, these disparities can now be resolved by using so called intrinsic metrics instead of the combinatorial graph distance. In this article we give an introduction to this topic and survey recent results in this direction. Specifically, we cover topics such as Liouville type theorems for harmonic functions, essential selfadjointness, stochastic completeness and upper escape rates. Furthermore, we determine the spectrum as a set via solutions, discuss upper and lower spectral bounds by isoperimetric constants and volume growth and study p-independence of spectra under a volume growth assumption.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Spectral Inequalities for Quantum Graphs
next chapter Spectral Gap for Complete Graphs: Upper and Lower Estimates
Literature
1.
Alon, N., Milman, V.D.: λ 1, isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory B 38(1), 73–88 (1985)
2.
Arlotti, L., Banasiak, J.: Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss, J. Math. Anal. Appl. 293(2), 693–720 (2004)MathSciNetCrossRef
3.
Azencott, R.: Behavior of diffusion semi-groups at infinity. Bull. Soc. Math. France 102, 193–240 (1974)MathSciNet
4.
Barlow, M.T., Bass, R.F.: Stability of parabolic Harnack inequalities. Trans. Am. Math. Soc. 356, 1501–1533 (2004)MathSciNetCrossRef
5.
6.
Bauer, F., Keller, M., Wojciechowski, R.: Cheeger inequalities for unbounded graph Laplacians. J. Eur. Math. Soc. (JEMS) 17(2), 259–271 (2015)
7.
Boutet de Monvel, A., Lenz, D., Stollmann, P.: Schnol’s theorem for strongly local forms. Isr. J. Math. 173, 189–211 (2009)
8.
9.
Brooks, R.: Spectral geometry and the Cheeger constant, expanding graphs. In: Proceedings of DIMACS Workshop, American Mathematical Society, pp. 5–19 (1993)
10.
Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969). pp. 195–199. Princeton University Press, Princeton (1970)
11.
Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12, 401–414 (1973)
12.
Colin de Verdière, Y., Torki-Hamza, N., Truc, F.: Essential self-adjointness for combinatorial Schrödinger operators II–metrically non complete graphs. Math. Phys. Anal. Geom. 14(1), 21–38 (2011)
13.
Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284(2), 787–794 (1984)MathSciNetCrossRef
14.
Dodziuk, J., Kendall, W. S.: Combinatorial Laplacians and isoperimetric inequality. In: From Local Times to Global Geometry, Control and Physics. Pitman Research Notes in Mathematics Series, vol. 150, pp. 68–74. Longman Scientific & Technical, Harlow (1986)
15.
Dodziuk, J., Karp, L.: Spectral and function theory for combinatorial Laplacians. In: Geometry of Random Motion (Ithaca, N.Y., 1987). Contemporary Mathematics, vol. 73, pp. 25–40. American Mathematical Society, Providence, RI (1988)
16.
Feller, W.: On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. Math. 65 (2), 527–570 (1957)MathSciNetCrossRef
17.
Feller, W.: Notes to my paper On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. Math. 68(2), 735–736 (1958)MathSciNetCrossRef
18.
Folz, M.: Gaussian upper bounds for heat kernels of continuous time simple random walks. Electron. J. Probab. 16(62), 1693–1722 (2011)MathSciNet
19.
20.
21.
Frank, L., Lenz, D., Wingert, D.: Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. J. Funct. Anal. 266(8), 4765–4808 (2014)MathSciNetCrossRef
22.
Fujiwara, K.: Growth and the spectrum of the Laplacian of an infinite graph. Tohoku Math. J. (2) 48(2), 293–302 (1996)
23.
24.
Fukushima, M., Ōshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. In: de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter, Berlin (1994)
25.
Gaffney, M.P.: The harmonic operator for exterior differential forms. Proc. Natl. Acad. Sci. U. S. A. 37, 48–50 (1951)MathSciNetCrossRef
26.
Gaffney, M.P.: A special Stokes’s theorem for complete Riemannian manifolds. Ann. Math. (2) 60, 140–145 (1954)
27.
Georgakopoulos, A., Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.: Graphs of finite measure. arXiv:1309.3501 to appear in J. Math. Pures Appl.
28.
Grigor’yan, A.: Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290(3), 534–537 (1986)MathSciNet
29.
Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36(2), 135–249 (1999)CrossRef
30.
Grigor’yan, A.: Escape rate of Brownian motion on Riemannian manifolds. Appl. Anal. 71(1–4), 63–89 (1999)MathSciNet
31.
Grigor’yan, A.: Analysis on Graphs. Lecture Notes. University Bielefeld, Bielefeld (2009)
32.
Grigor’yan, A., Hsu, E.P.: Volume growth and escape rate of Brownian motion on a Cartan-Hadamard manifold. In: Sobolev Spaces in Mathematics. II. International Mathematics Series, vol. 9, pp. 209–225. Springer, New York (2009)
33.
Grigor’yan, A., Huang, X., Masamune, J.: On stochastic completeness of jump processes. Math. Z. 271(3–4), 1211–1239 (2012)MathSciNetCrossRef
34.
Güneysu, B., Keller, M., Schmidt, M.: A Feynman-Kac-Itô formula for magnetic Schrödinger operators on graphs. arXiv:1301.1304 (preprint 2013)
35.
Haeseler, S., Keller, M.: Generalized solutions and spectrum for Dirichlet forms on graphs. In: Lenz, D., Sobieczky, F., Woess, W. (eds.): Random Walks, Boundaries and Spectra. Progress in Probability, vol. 64, pp. 181–199. Birkhäuser Verlag, Basel (2011)
36.
Haeseler, S., Keller, M., Lenz, D., Wojciechowski, R.K.: Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory 2(4), 397–432 (2012)MathSciNetCrossRef
37.
Haeseler, S., Keller, M., Wojciechowski, R.K.: Volume growth and bounds for the essential spectrum for Dirichlet forms. J. Lond. Math. Soc. 88(3), 883–898 (2013)MathSciNetCrossRef
38.
Hempel, R., Voigt, J.: The spectrum of a Schrödinger operator in \(L_{p}(\mathbb{R}^{\nu })\) is p-independent. Commun. Math. Phys. 104, 243–250 (1986)MathSciNetCrossRef
39.
Holopainen, I., Soardi, P.M.: A strong Liouville theorem for p-harmonic functions on graphs. Ann. Acad. Sci. Fenn. Math. 22, 205–226 (1997)MathSciNet
40.
Hsu, E.P., Qin, G.: Volume growth and escape rate of Brownian motion on a complete Riemannian manifold. Ann. Probab. 38(4), 1570–1582 (2010)MathSciNetCrossRef
41.
Hua, B., Jost, J.: l q harmonic functions on graphs. Isr. J. Math. 202(1), 475–490 (2014)
42.
Hua, B., Keller, M.: Harmonic functions of general graph Laplacians. Partial Differential Equations. Calc. Var 51(1–2), 343–362 (2014)MathSciNetCrossRef
43.
Huang, X.: On stochastic completeness of weighted graphs, Ph.D. Thesis (2011)
44.
Huang, X.: A note on the volume growth criterion for stochastic completeness of weighted graphs. Potential Anal. 40(2), 117–142 (2014)MathSciNetCrossRef
45.
Huang, X.: Escape rate of Markov chains on infinite graphs. J. Theoret. Probab. Doi: 10.1007/s10959-012-0456-x (to appear)
46.
47.
Huang, X., Shiozawa, Y.: Upper escape rate of Markov chains on weighted graphs. Stoch. Process. Appl. 124(1), 317–347 (2014)MathSciNetCrossRef
48.
Huang, X., Keller, M., Masamune, J., Wojciechowski, R.: A note on self-adjoint extensions of the Laplacian on weighted graphs. J. Funct. Anal. 265(8), 1556–1578 (2013)MathSciNetCrossRef
49.
50.
51.
Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. (Crelle’s J.) 666, 189–223 (2012)
52.
Keller, M., Lenz, D.: Unbounded Laplacians on graphs: basic spectral properties and the heat equation. Math. Model. Natl. Phenom. 5 (4), 198–224 (2010)MathSciNetCrossRef
53.
Keller, M., Lenz, D., Schmidt, M., Wirth, M.: Diffusion determines the recurrent graph. Adv. Math. 269, 364–398 (2015)MathSciNetCrossRef
54.
Keller, M., Lenz, D., R. Wojciechowski, K.: Volume growth, spectrum and stochastic completeness of infinite graphs. Math. Z. 274 (3–4), 905–932 (2013)
55.
56.
57.
Masamune, J.: A Liouville property and its application to the Laplacian of an infinite graph. In: Spectral Analysis in Geometry and Number Theory. Contemporary of Mathematics, vol. 484, pp. 103–115. American Mathematical Society, Providence, RI (2009)
58.
Masamune, J., Uemura, T.: Conservation property of symmetric jump processes. Ann. Inst. Henri Poincaré Probab. Stat. 47(3), 650–662 (2011)MathSciNetCrossRef
59.
Masamune, J., Uemura, T., Wang, J.: On the conservativeness and the recurrence of symmetric jump-diffusions. J. Funct. Anal. 263(12), 3984–4008 (2012)MathSciNetCrossRef
60.
Milatovic, O.: Essential self-adjointness of discrete magnetic Schrödinger operators on locally finite graphs. Integr. Equ. Oper. Theory 71, 13–27 (2011)MathSciNetCrossRef
61.
Milatovic, O.: A Sears-type self-adjointness result for discrete magnetic Schrödinger operators. J. Math. Anal. Appl. 396, 801–809 (2012)MathSciNetCrossRef
62.
Milatovic, O., Truc, F.: Self-adjoint extensions of discrete magnetic Schrödinger operators. Ann. Henri Poincaré 15, 917–936 (2014)MathSciNetCrossRef
63.
Mokhtar-Kharroubi, M., Voigt, J.: On honesty of perturbed substochastic C 0-semigroups in L 1-spaces. J. Oper. Theory 64(1), 131–147 (2010)MathSciNet
64.
Ohno, Y., Urakawa, H.: On the first eigenvalue of the combinatorial Laplacian for a graph. Interdiscip. Inform. Sci. 1(1), 33–46 (1994)MathSciNet
65.
Ouyang, S.: Volume growth, comparison theorem and escape rate of diffusion process. arXiv:1310.3996 (Preprint 2013)
66.
Pinsky, M.A.: Mean exit time of a diffusion process from a small sphere. Proc. Am. Math. Soc. 93(1), 157–158 (1985)MathSciNetCrossRef
67.
Reuter, G.E.H.: Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97, 1–46 (1957)
68.
69.
Roelcke, W.: Über den Laplace-Operator auf Riemannschen Mannigfaltigkeiten mit diskontinuierlichen Gruppen. Math. Nachr. 21, 131–149 (1960)MathSciNetCrossRef
70.
Schmidt, M.: Global properties of Dirichlet forms on discrete spaces. Diplomarbeit 2011. arXiv:1201.3474v2
71.
Shnol’, I. E.: On the behaviour of the eigenfunctions of Schrödinger’s eqution. Mat. Sb. 42, 273–286 (1957). Erratum 46(88), 259 (1957)
72.
73.
Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7(3), 447–526 (1982)CrossRef
74.
Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52(1), 48–79 (1983)MathSciNetCrossRef
75.
Sturm, K.-T.: On the lp-spectrum of uniformly elliptic operators on riemannian manifolds. J. Funct. Anal. 118, 442–453 (1993)MathSciNetCrossRef
76.
Sturm, K.-T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)
77.
Weidmann, J.: Linear operators in Hilbert spaces. In: Graduate Texts in Mathematics, vol. 68. Springer, New York (1980)
78.
Woess, W.: Random walks on infinite graphs and groups. In: Cambridge Tracts in Mathematics, vol. 138. Cambridge University Press, Cambridge (2000)
79.
Wojciechowski, R.K.: Stochastic completeness of graphs. ProQuest LLC, Ann Arbor, MI, Ph.D. Thesis, University of New York (2008)
80.
Wojciechowski, R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58, 1419–1441 (2009)MathSciNetCrossRef
81.
Wojciechowski, R.K.: Stochastically incomplete manifolds and graphs. In: Lenz, D., Sobieczky, F., Woess, W. (eds.) Random Walks, Boundaries and Spectra. Progress in Probability, vol. 64, pp. 163–179. Birkhäuser Verlag, Basel (2011)CrossRef
82.
Yau, S.T.: Some function-theoretic properties of complete Riemannian manifold and their application to geometry. Indiana Univ. Math. J. 25(7), 659–670 (1976)MathSciNetCrossRef
Metadata
Title
Intrinsic Metrics on Graphs: A Survey
Author
Matthias Keller
Copyright Year
2015
Book
Mathematical Technology of Networks

Print ISBN: 978-3-319-16618-6

Electronic ISBN: 978-3-319-16619-3

Copyright Year: 2015

DOI
https://doi.org/10.1007/978-3-319-16619-3_7

Premium Partner

    Image Credits