Homological eigenvalues of graph p-Laplacians

Abstract

Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph [Formula: see text]-Laplacian [Formula: see text], which allows us to analyze and classify non-variational eigenvalues. We show the stability of homological eigenvalues, and we prove that for any homological eigenvalue [Formula: see text], the function [Formula: see text] is locally increasing, while the function [Formula: see text] is locally decreasing. As a special class of homological eigenvalues, the min–max eigenvalues [Formula: see text] are locally Lipschitz continuous with respect to [Formula: see text]. We also establish the monotonicity of [Formula: see text] and [Formula: see text] with respect to [Formula: see text]. These results systematically establish a refined analysis of [Formula: see text]-eigenvalues for varying [Formula: see text], which lead to several applications, including: (1) settle an open problem by Amghibech on the monotonicity of some function involving eigenvalues of [Formula: see text]-Laplacian with respect to [Formula: see text]; (2) resolve a question asking whether the third eigenvalue of graph [Formula: see text]-Laplacian is of min–max form; (3) refine the higher-order Cheeger inequalities for graph [Formula: see text]-Laplacians by Tudisco and Hein, and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan to the [Formula: see text]-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.