Equivalent spectral theory for fundamental graph cut problems

Abstract: We introduce and develop equivalent spectral graph theory for several fundamental graph cut problems including maxcut, mincut, Cheeger cut, anti-Cheeger cut, dual Cheeger problem and their useful variants. The finer structure of the eigenvectors, the Courant nodal domain theorem and the graphic feature of eigenvalues are studied systematically in the setting of these new nonlinear eigenproblems. A specified strategy for achieving an equivalent eigenproblem is proposed for a general graph cut problem via the Dinkelbach scheme. We also establish several multi-way inequalities relating the variational eigenvalues and min-max $k$-cut problems, which closely resembles the higher-order Cheeger inequality on graph 1-Laplacian.