Essential self-adjointness for combinatorial Schr\"odinger operators II- Metrically non complete graphs

We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schr\"odinger operators in the metrically non complete case.Comment: Revisited version: Ognjen Milatovic wrote to us that he had discovered a gap in the proof of theorem 4.2 of our paper. As a consequence we propose to make an additional assumption (regularity property of the graph) to this theorem. A new subsection (4.1) is devoted to the study of this property and some details have been changed in the proof of theorem 4.