Higher eigenvalues of graphs
Name
Kelner-2009-Higher eigenvalues of graphs.pdf
Size
285.83 KB
Format
Adobe PDF
Checksum (MD5)
a5743110a799835397ab947598e2afa9
Author(s)
Price, Gregory N.
Lee, James R.
Teng, Shang-Hua
Kelner, Jonathan Adam
Date Issued
March 2010
Journal
50th Annual IEEE Symposium on Foundations of Computer Science, 2009. FOCS '09
Publisher
Institute of Electrical and Electronics Engineers
Citation
Kelner, J.A. et al. “Higher Eigenvalues of Graphs.” Foundations of Computer Science, 2009. FOCS '09. 50th Annual IEEE Symposium on. 2009. 735-744. ©2009 Institute of Electrical and Electronics Engineers.
Version
Final published version
Abstract
We present a general method for proving upper bounds on the eigenvalues of the graph Laplacian. In particular, we show that for any positive integer k, the kth smallest eigenvalue of the Laplacian on a 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 planar grids. We also extend this spectral result to graphs with bounded genus, graphs which forbid fixed minors, and other natural families. Previously, such spectral upper bounds were only known for k = 2, i.e. for the Fiedler value of these graphs. In addition, our result yields a new, combinatorial proof of the celebrated result of Korevaar in differential geometry.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1109/FOCS.2009.69
