  We study Laplacians on graphs and networks via regular Dirichlet forms. We
give a sufficient geometric condition for essential selfadjointness and
explicitly determine the generators of the associated semigroups on all
$\ell^p$, $1\leq p < \infty$, in this case. We characterize stochastic
completeness thereby generalizing all earlier corresponding results for graph
Laplacians. Finally, we study how stochastic completeness of a subgraph is
related to stochastic completeness of the whole graph.
