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The Graph \(\boldsymbol{\infty }\)-Laplacian Eigenvalue Problem.

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Title: The Graph \(\boldsymbol{\infty }\)-Laplacian Eigenvalue Problem.
Authors: Deidda, Piero1 (AUTHOR) piero.deidda@sns.it, Burger, Martin2 (AUTHOR) martin.burger@desy.de, Putti, Mario3 (AUTHOR) mario.putti@unipd.it, Tudisco, Francesco4 (AUTHOR) f.tudisco@ed.ac.uk
Source: SIAM Journal on Mathematical Analysis. 2026, Vol. 58 Issue 1, p1-34. 34p.
Subjects: Eigenvalues, Laplacian operator, Rayleigh quotient, Eigenfunctions, Graph theory, Variational approach (Mathematics), Packing problem (Mathematics)
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 establishes 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 from both above and 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. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
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 establishes 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 from both above and 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. [ABSTRACT FROM AUTHOR]
ISSN:00361410
DOI:10.1137/24M1705342