Graph \({p}\)-Laplacian Eigenpairs as Saddle Points of a Family of Spectral Energy Functions

We address the problem of computing the graph p-Laplacian eigenpairs for p \in (2, \infty). We propose a reformulation of the graph p-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and discuss theoretical and computational advantages. We provide a correspondence between p-Laplacian eigenpairs and linear eigenpairs of a constrained generalized weighted Laplacian eigenvalue problem. As a result, we can assign an index to any p-Laplacian eigenpair that matches the Morse index of the p-Rayleigh quotient evaluated at the eigenfunction. In the second part of the paper, we introduce a class of spectral energy functions that depend on edge and node weights. We prove that differentiable saddle points of the kth energy function correspond to p-Laplacian eigenpairs having index equal to k. Moreover, the first energy function is proved to possess a unique saddle point which corresponds to the unique first p-Laplacian eigenpair. Finally, we develop novel gradient-based numerical methods suited to compute p-Laplacian eigenpairs for any p \in (2, \infty) and present some experiments.