Self-adjoint and Markovian extensions of infinite quantum graphs.

We investigate the relationship between one of the classical notions of boundaries for infinite graphs, , and self-adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self-adjointness of the Gaffney Laplacian - the underlying metric graph does not have finite volume ends.