Spectrum of the Signless 1-Laplacian and the Dual Cheeger Constant on Graphs

Parallel to the signless Laplacian spectral theory, we introduce and develop the nonlinear spectral theory of signless 1-Laplacian on graphs. Again, the first eigenvalue μ1 of the signless 1-Laplacian precisely characterizes the bipartiteness of a graph and naturally connects to the maxcut problem. However, the dual Cheeger constant h+, which has only some upper and lower bounds in the Laplacian spectral theory, is proved to be 1− μ1 . The structure of the eigenvectors and the graphic feature of eigenvalues are also studied. The Courant nodal domain theorem for graphs is extended to the signless 1-Laplacian. A set-pair version of the Lovász extension, which aims at the equivalence between discrete combination optimization and continuous function optimization, is established to recover the relationship h+ = 1−μ1 . A local analysis of the associated functional yields an inverse power method to determine h+ and then produces an efficient implementation of the recursive spectral cut algorithm for the maxcut problem.