Metric uniformization and spectral bounds for graphs
Name
Kelner_Metric Uniformization.pdf
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272.15 KB
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Adobe PDF
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Author(s)
Kelner, Jonathan Adam
Lee, James R.
Price, Gregory N.
Teng, Shang-Hua
Date Issued
August 2011
Journal
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
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/s00039-011-0132-9
