The graph $\infty$-Laplacian eigenvalue problem

TL;DR

This paper studies the $ abla$-Laplacian eigenvalue problem on graphs, extending continuous results to discrete graphs, introducing generalized eigenpairs, and analyzing their properties and bounds related to graph packing radii.

Contribution

It extends the theory of $ abla$-Laplacian eigenvalues to graphs, introduces generalized eigenpairs, and establishes bounds and relationships with graph packing radii.

Findings

Limit of $p$-Laplacian eigenvalues bounds by packing radius.

Generalized eigenvalues match limits of variational $p$-Laplacian eigenvalues.

Solutions to the limit eigenvalue equation relate to generalized eigenpairs.

Abstract

We analyze various formulations of the $\infty$-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph $\infty$-eigenpairs arising as limits of $p$-Laplacian eigenpairs, extending key results from the continuous setting to the discrete domain. We prove that every limit of $p$-Laplacian eigenpair, for $p$ going to $\infty$, satisfies a limit eigenvalue equation and establish that the corresponding eigenvalue can be bounded from below by the packing radius of the graph, indexed by the number of nodal domains induced by the eigenfunction. Additionally, we show that the limits, for $p$ going to $\infty$, of the variational $p$-Laplacian eigenvalues are bounded both from above and from below by the packing radii, achieving equality for the smallest two variational eigenvalues and corresponding packing radii of the graph. In the second part of the paper, we introduce generalized $\infty$-Laplacian eigenpairs as generalized critical points and values of the $\infty$-Rayleigh quotient. We prove that the generalized variational $\infty$-eigenvalues equal the limit of the $p$-Laplacian variational eigenvalues and so satisfy the same upper bounds in terms of packing radii. Finally, we establish that any solution to the limit eigenvalue equation is also a generalized eigenpair, while any generalized eigenpair satisfies the limit eigenvalue equation on a suitable subgraph.