Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions

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 �nite 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 how these may dier from each 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 posi- tivity improving and present some basic estimates on its long term behavior. We also characterize boundedness of the Laplacian.