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 1Laplacian. 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 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 non-zero eigenvalue for connected graphs. An algorithum of Cheegers cut, based on the above characterization is also studied.