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 the Ambartsumian theorem is proven under the assumption that the vertex conditions are asymptotically properly connecting and asymptotically standard. By constructing explicit counterexamples, it is shown that the geometric Ambartsumian theorem does not hold in general without additional assumptions on the vertex conditions.