2024/05/11 by Piero Deidda, Deidda, Piero, Nicola Segala +3 · 1 citation
Mathematics · #FOS: Mathematics #Graph theory and applications #Numerical Analysis (math.NA) #Spectral Theory in Mathematical Physics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2405.07056
openalex publication_date 2024/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We address the problem of computing the graph p-Laplacian eigenpairs for p∈ (2,∞). 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 eigenpair 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 k-th 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∈(2,∞) and present some experiments.