  We present a method for proving upper bounds on the eigenvalues of the graph
Laplacian. A main step involves choosing an appropriate "Riemannian" metric to
uniformize the geometry of the graph. In many interesting cases, the existence
of such a metric is shown by examining the combinatorics of special types of
flows. This involves proving new inequalities on the crossing number of graphs.
  In particular, we use our method to show that for any positive integer k, the
kth smallest eigenvalue of the Laplacian on an n-vertex, bounded-degree planar
graph is O(k/n). This bound is asymptotically tight for every k, as it is
easily seen to be achieved for square planar grids. We also extend this
spectral result to graphs with bounded genus, and graphs which forbid fixed
minors. Previously, such spectral upper bounds were only known for the case
k=2.
