Journal of Spectral Theory

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Volume 2, Issue 4, 2012, pp. 397–432
DOI: 10.4171/JST/35

Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions

Sebastian Haeseler[1], Matthias Keller[2], Daniel Lenz[3] and Radosław K. Wojciechowski[4]

(1) Mathematisches Institut, Friedrich-Schiller-Universität Jena, 07743, Jena, Germany
(2) Einstein Institute of Mathematics, The Hebrew University, 91904, Jerusalem, Israel
(3) Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany
(4) Department of Mathematics and Computer Science, York College of The City University of New York, NY 11451, Jamaica, USA

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.

Keywords: Laplacian on graphs, Dirichlet boundary conditions, Neumann boundary conditions, Markovian extension

Haeseler Sebastian, Keller Matthias, Lenz Daniel, Wojciechowski Radosław: Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions. J. Spectr. Theory 2 (2012), 397-432. doi: 10.4171/JST/35