JournalsjstVol. 2, No. 4pp. 397–432

Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions

  • Sebastian Haeseler

    Friedrich-Schiller-Universität Jena, Germany
  • Matthias Keller

    The Hebrew University, Jerusalem, Israel
  • H. Daniel Lenz

    Friedrich-Schiller-Universität Jena, Germany
  • Radosław K. Wojciechowski

    York College of The City University of New York, Jamaica, USA
Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions cover
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Abstract

We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard formal Laplacian and we can characterize the Dirichlet and Neumann Laplacians as the largest and smallest Markovian restrictions of the standard formal Laplacian. In the case of general graphs, this class contains the Dirichlet and Neumann Laplacians and we describe howthesemay differ fromeach other, characterize when they agree, and study connections to essential selfadjointness and stochastic completeness.

Finally, we study basic common features of all Laplacians associated to a graph. In particular, we characterize when the associated semigroup is positivity improving and present some basic estimates on its long term behavior. We also discuss some situations in which the Laplacian associated to a graph is unique and, in this context, characterize its boundedness.

Cite this article

Sebastian Haeseler, Matthias Keller, H. Daniel Lenz, Radosław K. Wojciechowski, Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory 2 (2012), no. 4, pp. 397–432

DOI 10.4171/JST/35