Graph $p$-Laplacian eigenpairs as saddle points of a family of spectral energy functions
<|reference_start|>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 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 $p\in(2,\infty)$ and present some experiments.<|reference_end|>
@article{deidda2024graph,
    title={Graph $p$-Laplacian eigenpairs as saddle points of a family of spectral
  energy functions},
    author={Piero Deidda, Nicola Segala, Mario Putti},
    journal={arXiv preprint arXiv:2405.07056},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.07056},
    primaryClass={math.NA cs.NA}
}