Maximal dissipative operators on metric graphs: Real eigenvalues and their multiplicities

Dissipative Schrödinger operators on metric graphs are discussed. Vertex conditions leading to maximal dissipative operators are characterised. The language of hypergraphs is introduced and used to determine possible spectral multiplicities of the self-adjoint reductions, which depends not only on the properties of the potential but on the topologic and geometric properties of the metric graph. This leads to the characterisation of all operators, not possessing any self-adjoint reduction, so-called completely non-self-adjoint operators, on compact metric graphs with delta couplings at the vertices.