Converse between essential self-adjointness and χ-completeness in graphs

Determine whether, in the graph case (n=1), essential self-adjointness of the 1-form Laplacian Delta_1 on a locally finite weighted graph implies global χ-completeness of the graph, i.e., the existence of an exhaustion by finite vertex sets together with plateau cut-off functions having uniformly bounded discrete energy.

Background

In the graph case (n=1), prior work (BGJ) established that χ-completeness holds if and only if a specific divergence condition on the offspring function is satisfied, and that this condition implies essential self-adjointness (ESA) of the 1-form Laplacian Delta_1.

However, BGJ explicitly remarked that the converse—whether ESA of Delta_1 implies χ-completeness—remains unresolved. The present paper provides higher-dimensional counterexamples showing ESA without χ-completeness for n ≥ 2, thereby resolving the question beyond graphs, but leaving the graph case itself as an open problem.