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 k th 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.

Authors

• Jonathan A. Kelner
• James R. Lee
• Gregory N. Price
• Shang-Hua Teng

Keywords

• Eigenvalues and eigenfunctions
• Laplace equations
• Transmission line matrix methods
• Upper bound
• Optimization methods
• Image segmentation
• Partitioning algorithms
• Very large scale integration
• Computer science
• Geometry
• Higher Eigenvalues
• Eigenvalues Of Graphs
• Differential Geometry
• Planar Graphs
• Laplacian Eigenvalues
• Cardinality
• Eigenvectors
• Undirected
• Lagrange Multiplier
• Convex Optimization
• Spectral Method
• Flow Problem
• Optimal Flow
• Spectral Algorithm
• Spectral Graph Theory
• Kind Of Flow
• Grid Graph