In this paper, we prove a variant of the Burger-Brooks transfer principle which, combined with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a function of their genus. More precisely, we show the existence of a universal constants C such that the k-th eigenvalue λ_k of the normalized Laplacian of a graph G of (geometric) genus g on n vertices satisfies λ_k ≤Cdmax(g+k) / kn where dmax denotes the maximum valence of vertices of the graph. This result is tight up to a change in the value of the constant C. We also use our transfer theorem to relate eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple graph models, and discuss an application to the mesh partitioning problem.