We study the uniqueness of self-adjoint and Markovian exten
sions of the Laplacian
on weighted graphs. We first show that, for locally finite grap
hs and a certain family of metrics,
completeness of the graph implies uniqueness of these exten
sions. Moreover, in the case when the
graph is not metrically complete and the Cauchy boundary has
finite capacity, we characterize
the uniqueness of the Markovian extensions
