Homological eigenvalues of graph $p$-Laplacians

TL;DR

This paper introduces homological eigenvalues for graph p-Laplacians, analyzing their stability, monotonicity, and continuity with respect to p, and applies these findings to solve open problems and extend Cheeger inequalities.

Contribution

It defines homological eigenvalues for graph p-Laplacians, proves their stability and monotonicity, and applies these results to solve open problems and improve inequalities.

Findings

Homological eigenvalues are stable and monotonic with respect to p.

Min-max eigenvalues are locally Lipschitz continuous in p.

Applications include solving open problems and refining Cheeger inequalities.

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 $p\mapsto p(2\lambda(\Delta_p))^{\frac1p}$ 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 $p(2\lambda_k(\Delta_p))^{\frac1p}$ 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.