Homological eigenvalues of graph $p$-Laplacians

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.