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In this paper, we prove a variant of the Burger-Brooks transfer principlewhich, combined with recent eigenvalue bounds for surfaces, allows to obtainupper bounds on the eigenvalues of graphs as a function of their genus. Moreprecisely, 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 ofvertices of the graph. This result is tight up to a change in the value of theconstant $C$, and improves recent results of Kelner, Lee, Price and Teng onbounded genus graphs.  To show that the transfer theorem might be of independent interest, we relateeigenvalues of the Laplacian on a metric graph to the eigenvalues of its simplegraph models, and discuss an application to the mesh partitioning problem,extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tangto arbitrary meshes.

A transfer principle and applications to eigenvalue estimates for graphs