  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 $\lambda_k^{nr}$ of the normalized Laplacian of a graph $G$
of (geometric) genus $g$ on $n$ vertices satisfies $$\lambda_k^{nr}(G) \leq C
\frac{d_{\max}(g+k)}{n},$$ where $d_{\max}$ denotes the maximum valence of
vertices of the graph. This result is tight up to a change in the value of the
constant $C$, and improves recent results of Kelner, Lee, Price and Teng on
bounded genus graphs.
  To show that the transfer theorem might be of independent interest, we 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,
extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang
to arbitrary meshes.
