Higher eigenvalues of graphs
Author(s)Price, Gregory N.; Lee, James R.; Teng, Shang-Hua; Kelner, Jonathan Adam
MetadataShow full item record
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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
50th Annual IEEE Symposium on Foundations of Computer Science, 2009. FOCS '09
Institute of Electrical and Electronics Engineers
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.
Final published version
INSPEC Accession Number: 11207105