  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 how these may differ 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
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.
