Eigenvalues of the discrete p-Laplacian via graph surgery

We develop a Hellmann–Feynman type perturbation theory for the discrete signed $p$-Laplacian and apply it to a parametrized perturbation by edge cuts. We show that the eigenvalues of the signed $p$-Laplacian can be characterized as citical values of the parameter-dependent eigenvalues of a simpler graph.