Nodal geometry of graphs on surfaces

Yong Lin; Gábor Lippner; Dan Mangoubi; Shing Tung Yau

doi:10.3934/dcds.2010.28.1291

Nodal geometry of graphs on surfaces

Yong Lin^*
, Gábor Lippner
, Dan Mangoubi
, Shing Tung Yau

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus g. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the nth Laplacian eigenvalue is at most 2 [6(n - 1) + 15(2g - 2)]². Our results hold for any Schrodinger operator, not just the Laplacian.

Original language	English
Pages (from-to)	1291-1298
Number of pages	8
Journal	Discrete and Continuous Dynamical Systems
Volume	28
Issue number	3
DOIs	https://doi.org/10.3934/dcds.2010.28.1291
State	Published - Nov 2010

Keywords

Genus
Multiplicity of eigenvalues
Nodal domain

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10.3934/dcds.2010.28.1291

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title = "Nodal geometry of graphs on surfaces",

abstract = "We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus g. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the nth Laplacian eigenvalue is at most 2 [6(n - 1) + 15(2g - 2)]2. Our results hold for any Schrodinger operator, not just the Laplacian.",

keywords = "Genus, Multiplicity of eigenvalues, Nodal domain",

author = "Yong Lin and G{\'a}bor Lippner and Dan Mangoubi and Yau, \{Shing Tung\}",

year = "2010",

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doi = "10.3934/dcds.2010.28.1291",

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AU - Mangoubi, Dan

AU - Yau, Shing Tung

PY - 2010/11

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N2 - We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus g. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the nth Laplacian eigenvalue is at most 2 [6(n - 1) + 15(2g - 2)]2. Our results hold for any Schrodinger operator, not just the Laplacian.

AB - We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus g. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the nth Laplacian eigenvalue is at most 2 [6(n - 1) + 15(2g - 2)]2. Our results hold for any Schrodinger operator, not just the Laplacian.

KW - Genus

KW - Multiplicity of eigenvalues

KW - Nodal domain

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JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

IS - 3

ER -

Nodal geometry of graphs on surfaces

Abstract

Keywords

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