Discrete nodal domain theorems

BrianDavies, E.; Gladwell, GrahamM.L.; Leydold, Josef; Stadler, Peter F.

doi:10.1016/s0024-3795(01)00313-5

Cited by 124 publications

(150 citation statements)

References 9 publications

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“…Sturm's oscillation theorem for systems in 1-d, and its extension by Courant to any dimension, establish the connection between the number of nodal domains and the spectrum: The number of nodal domains ν n of the n'th eigenfunction is bounded by n. (the eigenfunctions are arranged by increasing value of their eigenvalues). Courant's theorem for combinatorial graphs and for quantum graphs were proved in [25,22], respectively.…”

Section: Nodal Counts and The Resolution Of Isospectralitymentioning

confidence: 99%

Nodal domains on isospectral quantum graphs: the resolution of isospectrality?

Band¹,

Shapira²,

Smilansky³

2006

J. Phys. A: Math. Gen.

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Abstract. We present and discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in R 2 which were introduced recently in [1-5] all based on Sunada's construction of isospectral domains[6]. After presenting some of the properties of these graphs, we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity.

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Section: Nodal Counts and The Resolution Of Isospectralitymentioning

confidence: 99%

Nodal domains on isospectral quantum graphs: the resolution of isospectrality?

Band¹,

Shapira²,

Smilansky³

2006

J. Phys. A: Math. Gen.

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“…There are two groups of eigenvalues, one by solutions {µ 1 , µ 2 , · · · } of the system (2) with natural order, another by c 1 ≤ c 2 ≤ · · · ≤ c n via the minimax principle (6). It was then asked [2]: Is there any eigenvalue µ, which is not in the sequence: c 1 ≤ c 2 ≤ · · · ≤ c n ?…”

Section: Introductionmentioning

confidence: 99%

Nodal domains of eigenvectors for 1-Laplacian on graphs

Chang

Shao

Zhang

2017

Advances in Mathematics

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The eigenvectors for graph 1-Laplacian possess some sort of localization property: On one hand, any nodal domain of an eigenvector is again an eigenvector with the same eigenvalue; on the other hand, one can pack up an eigenvector for a new graph by several fundamental eigencomponents and modules with the same eigenvalue via few special techniques. The Courant nodal domain theorem for graphs is extended to graph 1-Laplacian for strong nodal domains, but for weak nodal domains it is false. The notion of algebraic multiplicity is introduced in order to provide a more precise estimate of the number of independent eigenvectors. A positive answer is given to a question raised

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“…Of course, such a set of vertices is not "bounded" by "nodes"; it is merely "bounded" by vertices of the opposite loose sign. A more appropriate name for such an entity would thus appear to be sign graph [8]. We nevertheless use here the established terminology from the manifold case.…”

Section: Introduction the Eigenfunctions Of Elliptic Differential Eqmentioning

confidence: 99%

“…In general, generalized graph Laplacians satisfy an analog of Courant's Nodal Domain Theorem [8]. As in the case of manifolds, it is of interest to consider special classes of graphs for which stronger and more detailed results, e.g.…”

Section: Introduction the Eigenfunctions Of Elliptic Differential Eqmentioning

confidence: 99%

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Nodal domain theorems and bipartite subgraphs

Bıyıkoğlu¹,

Leydold²,

Stadler³

2005

ELA

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Abstract. The Discrete Nodal Domain Theorem states that an eigenfunction of the k-th largest eigenvalue of a generalized graph Laplacian has at most k (weak) nodal domains. The number of strong nodal domains is shown not to exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor.

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Discrete nodal domain theorems

Cited by 124 publications

References 9 publications

Nodal domains on isospectral quantum graphs: the resolution of isospectrality?

Nodal domains on isospectral quantum graphs: the resolution of isospectrality?

Nodal domains of eigenvectors for 1-Laplacian on graphs

Nodal domain theorems and bipartite subgraphs

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