Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law

We study self-adjoint extensions of a second order differential operator of Sturm-Liouville type on a graph. We relate self-adjointness of the operator to the existence of non-complete trajectories of the Hamiltonian vector field defined by its principal symbol outside the vertices. We define Kirchhoff conditions at the vertices which guarantee a self-adjoint extension analogous to the case of quantum graphs. The singular vertices may be interpreted as introducing a singular potential at those points. We also establish a Weyl’s law for the spectrum asymptotics.