Metric uniformization and spectral bounds for graphs
Author(s)
Kelner, Jonathan Adam; Lee, James R.; Price, Gregory N.; Teng, Shang-Hua
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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 [superscript 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.
Date issued
2011-08Department
Massachusetts Institute of Technology. Department of MathematicsJournal
Geometric and Functional Analysis
Publisher
Springer Science + Business Media B.V.
Citation
Kelner, Jonathan A. et al. “Metric Uniformization and Spectral Bounds for Graphs.” Geometric and Functional Analysis 21.5 (2011): 1117–1143. Web.
Version: Author's final manuscript
ISSN
1016-443X
1420-8970