  Formal Laplace operators are analyzed for a large class of resistance
networks with vertex weights. The graphs are completed with respect to the
minimal resistance path metric. Compactness and a novel connectivity hypothesis
for the completed graphs play an essential role. A version of the Dirichlet
problem is solved. Self adjoint Laplace operators and the probability
semigroups they generate are constructed using reflecting and absorbing
conditions on subsets of the graph boundary.
