Spectrum of the 1‐Laplacian and Cheeger's Constant on Graphs

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1 - Laplacian Δ1. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first nonzero Δ1 eigenvalue for connected graphs.

[1]  Matthias Hein,et al.  Spectral clustering based on the graph p-Laplacian , 2009, ICML '09.

[2]  Martin Schechter,et al.  Critical point theory and its applications , 2006 .

[3]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[4]  B. Kawohl,et al.  DIRICHLET PROBLEMS FOR THE 1-LAPLACE OPERATOR, INCLUDING THE EIGENVALUE PROBLEM , 2007 .

[5]  Kung-Ching Chang.  THE SPECTRUM OF THE 1-LAPLACE OPERATOR , 2009 .

[6]  P. Rabinowitz.  Minimax methods in critical point theory with applications to differential equations , 1986 .

[7]  Matthias Hein,et al.  An Inverse Power Method for Nonlinear Eigenproblems with Applications in 1-Spectral Clustering and Sparse PCA , 2010, NIPS.

[8]  Peter F. Stadler,et al.  Laplacian Eigenvectors of Graphs , 2007 .

[9]  Bernhard Kawohl,et al.  Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant , 2003 .

[10]  Marco Degiovanni,et al.  Deformation properties for continuous functionals and critical point theory , 1993 .

[11]  Kung-Ching Chang,et al.  Variational methods for non-differentiable functionals and their applications to partial differential equations , 1981 .

[12]  J. Cheeger.  A lower bound for the smallest eigenvalue of the Laplacian , 1969 .

[13]  Matthias Hein,et al.  Beyond Spectral Clustering - Tight Relaxations of Balanced Graph Cuts , 2011, NIPS.

[14]  Saïd Amghibech,et al.  Eigenvalues of the Discrete p-Laplacian for Graphs , 2003, Ars Comb..

[15]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[16]  Xavier Bresson,et al.  Total Variation, Cheeger Cuts , 2010, ICML.