A note on self-adjoint extensions of the Laplacian on weighted graphs
  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. 
Xueping Huang, Matthias Keller, Jun Masamune, and Rados{\l}aw K.   Wojciechowski
math.FA