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 k th 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.
For example, Kelner et al. REF show that for n-vertex, bounded-degree planar graphs, one has that the kth smallest eigenvalue satisfies λ k = O(k/n).
Metric uniformization and spectral bounds for graphs
ArXiv
ArXiv
Mathematics Computer Science