In this paper we study Schrodinger operators with absolutely integrable potentials on metric graphs. Uniform bounds-i.e. depending only on the graph and the potential-on the difference between the eigenvalues of the Laplace and Schrodinger operators are obtained. This in turn allows us to prove an extension of the classical Ambartsumian Theorem which was originally proven for Schrodinger operators with Neumann conditions on an interval. We also extend a previous result relating the spectrum of a Schrodinger operator to the Euler characteristic of the underlying metric graph.