  We study the uniqueness of self-adjoint and Markovian extensions of the
Laplacian on weighted graphs. We first show that, for locally finite graphs and
a certain family of metrics, completeness of the graph implies uniqueness of
these extensions. 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.
