Higher Eigenvalues of Graphs

— 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.

• Type: Conference
• Year: 2009
• Where: FOCS
• Authors: Jonathan A. Kelner, James R. Lee, Gregory N. Price, Shang-Hua Teng
• Keywords: Bounded-degree Planar Graph, FOCS 2009, Kth Smallest Eigenvalue, Theoretical Computer Science, Upper Bounds