Homological eigenvalues of graph $p$-Laplacians

Abstract

Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph $p$-Laplacian ${\Delta }_p$, which allows us to analyse and classify non-variational eigenvalues. We show the stability of homological eigenvalues, and we prove that for any homological eigenvalue $\lambda ({\Delta }_p)$, the function $p\mapsto p(2\lambda ({\Delta }_p){)}^{\frac{1}{p}}$ is locally increasing, while the function $p\mapsto 2^{-p}\lambda ({\Delta }_p)$ is locally decreasing. As a special class of homological eigenvalues, the min-max eigenvalues ${\lambda }_1({\Delta }_p)$, $\cdots$, ${\lambda }_k({\Delta }_p)$, $\cdots$, are locally Lipschitz continuous with respect to $p\in [1,+\infty )$. We also establish the monotonicity of $p(2{\lambda }_k({\Delta }_p){)}^{\frac{1}{p}}$ and $2^{-p}{\lambda }_k({\Delta }_p)$ with respect to $p\in [1,+\infty )$. These results systematically establish a refined analysis of ${\Delta }_p$-eigenvalues for varying $p$, which lead to several applications, including: (1) settle an open problem by Amghibech on the monotonicity of some function involving eigenvalues of $p$-Laplacian with respect to $p$; (2) resolve a question asking whether the third eigenvalue of graph $p$-Laplacian is of min-max form; (3) refine the higher order Cheeger inequalities for graph $p$-Laplacians by Tudisco and Hein, and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan to the $p$-Laplacian case. Furthermore, for the 1-Laplacian case, we characterize the homological eigenvalues and min-max eigenvalues from the perspective of topological combinatorics, where our idea is similar to the authors' work on discrete Morse theory.

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References in corpus (4)

• Petals and Books: The largest Laplacian spectral gap from 1
• Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures
• Discrete-to-Continuous Extensions: piecewise multilinear extension, min-max theory and spectral theory
• Discrete-to-Continuous Extensions: Lovász extension, optimizations and eigenvalue problems