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 ℓp, 1 ≦ p < ∞, 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.
