  We give two characterizations for the essential self-adjointness of the
weighted Laplacian on birth-death chains. The first involves the edge weights
and vertex measure and is classically known; however, we give another proof
using stability results, limit point-limit circle theory and the connection
between essential self-adjointness and harmonic functions. The second
characterization involves a new notion of capacity. Furthermore, we also
analyze the essential self-adjointness of Schr\"odinger operators, use the
characterizations for birth-death chains and stability results to characterize
essential self-adjointness for star-like graphs, and give some connections to
the $\ell^2$-Liouville property.
