Homological eigenvalues of graph p-Laplacians.

Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph p -Laplacian Δ p , 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 λ (Δ p) , the function p ↦ p (2 λ (Δ p)) 1 p is locally increasing, while the function p ↦ 2 − p λ (Δ p) is locally decreasing. As a special class of homological eigenvalues, the min–max eigenvalues λ 1 (Δ p) , ... , λ k (Δ p) , ... , are locally Lipschitz continuous with respect to p ∈ [ 1 , + ∞). We also establish the monotonicity of p (2 λ k (Δ p)) 1 p and 2 − p λ k (Δ p) with respect to p ∈ [ 1 , + ∞). These results systematically establish a refined analysis of Δ 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. [ABSTRACT FROM AUTHOR]