Work

Graph \({p}\)-Laplacian Eigenpairs as Saddle Points of a Family of Spectral Energy Functions

Year: 2025

Type: article

Abstract: . We address the problem of computing the graph [Formula: see text]-Laplacian eigenpairs for [Formula: see text]. We propose a reformulation of the graph [Formula: see text]-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and discuss theoretical and computational advantages. We provide a correspondence between [Formula: see text]-Laplacian eigenpairs and linear eigenpairs of a constrained generalized weighted Laplacian eigenvalue problem. As a result, we can assign an index to any [Formula: see text]-Laplacian eigenpair that matches the Morse index of the [Formula: see text]-Rayleigh quotient evaluated at the eigenfunction. In the second part of the paper, we introduce a class of spectral energy functions that depend on edge and node weights. We prove that differentiable saddle points of the [Formula: see text]th energy function correspond to [Formula: see text]-Laplacian eigenpairs having index equal to [Formula: see text]. Moreover, the first energy function is proved to possess a unique saddle point which corresponds to the unique first [Formula: see text]-Laplacian eigenpair. Finally, we develop novel gradient-based numerical methods suited to compute [Formula: see text]-Laplacian eigenpairs for any [Formula: see text] and present some experiments.

Source: SIAM Journal on Matrix Analysis and Applications

DOI: 10.1137/24m1664769

Authors: Piero Deidda, Nicola Segala, Mario Putti

Institutions: Scuola Normale Superiore, Gran Sasso Science Institute, University of Padua

Language: English

Topic: Graph theory and applications

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Istituto Nazionale di Alta Matematica "Francesco Severi", European Commission, Ministero dell'Università e della Ricerca, Gruppo Nazionale per il Calcolo Scientifico

Award: G1374306608