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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]
Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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Header DbId: egs
DbLabel: Engineering Source
An: 191988909
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PubTypeId: academicJournal
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  Data: The Graph \(\boldsymbol{\infty }\)-Laplacian Eigenvalue Problem.
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  Data: 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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  Data: <i>Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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      – Type: doi
        Value: 10.1137/24M1705342
    Languages:
      – Code: eng
        Text: English
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        PageCount: 34
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    Subjects:
      – SubjectFull: Eigenvalues
        Type: general
      – SubjectFull: Laplacian operator
        Type: general
      – SubjectFull: Rayleigh quotient
        Type: general
      – SubjectFull: Eigenfunctions
        Type: general
      – SubjectFull: Graph theory
        Type: general
      – SubjectFull: Variational approach (Mathematics)
        Type: general
      – SubjectFull: Packing problem (Mathematics)
        Type: general
    Titles:
      – TitleFull: The Graph \(\boldsymbol{\infty }\)-Laplacian Eigenvalue Problem.
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          Name:
            NameFull: Deidda, Piero
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            NameFull: Burger, Martin
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            NameFull: Putti, Mario
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            NameFull: Tudisco, Francesco
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          Dates:
            – D: 01
              M: 01
              Text: 2026
              Type: published
              Y: 2026
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              Value: 58
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            – TitleFull: SIAM Journal on Mathematical Analysis
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