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.

翻译：我们研究$p\in (2,\infty)$情形下图$p$-拉普拉斯算子特征对的计算问题。提出将图$p$-拉普拉斯算子特征值问题重新表述为带约束的加权拉普拉斯算子特征值问题，并讨论其理论与计算优势。我们建立了$p$-拉普拉斯算子特征对与约束广义加权拉普拉斯算子特征值问题的线性特征对之间的对应关系。因此，可以给任意$p$-拉普拉斯算子特征对赋予一个指标，该指标与在特征函数处评估的$p$-瑞利商的莫尔斯指标相匹配。在论文第二部分，我们引入一类依赖于边权重和节点权重的谱能量函数。证明第$k$个能量函数的可微鞍点对应于指标等于$k$的$p$-拉普拉斯算子特征对。此外，证明第一个能量函数具有唯一的鞍点，该鞍点对应于唯一的第一$p$-拉普拉斯算子特征对。最后，我们开发了适用于计算任意$p\in(2,\infty)$情形下图$p$-拉普拉斯算子特征对的基于梯度的新型数值方法，并给出若干实验。