Essential self-adjointness and the $L^2$-Liouville property

We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Greens function and when it gives a non-constant harmonic function which is square integrable.