Schrödinger operators on metric graphs with general vertex conditions are studied. Explicit spectral asymptotics is derived in terms of the spectrum of reference Laplacians. A geometric version of Ambartsumian theorem is proven under the assumptionthat the vertex conditions are asymptotically properly connecting and asymptotically standard. By constructing explicit counterexamples it is shown that the geometric Albartsumian theorem does not hold in general without additional assumptions on the vertex conditions.