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We study the uniqueness of self-adjoint and Markovian extensions of theLaplacian on weighted graphs. We first show that, for locally finite graphs anda certain family of metrics, completeness of the graph implies uniqueness ofthese extensions. Moreover, in the case when the graph is not metricallycomplete and the Cauchy boundary has finite capacity, we characterize theuniqueness of the Markovian extensions.

A note on self-adjoint extensions of the Laplacian on weighted graphs