  The Krein-von Neumann extension is studied for Schr\"odinger operators on
metric graphs. Among other things, its vertex conditions are expressed
explicitly, and its relation to other self-adjoint vertex conditions (e.g.
continuity-Kirchhoff) is explored. A variational characterisation for its
positive eigenvalues is obtained. Based on this, the behaviour of its
eigenvalues under perturbations of the metric graph is investigated, and
so-called surgery principles are established. Moreover, isoperimetric
eigenvalue inequalities are obtained.
