Nodal decompositions of graphs

A nodal domain of a function is a maximally connected subset of the domain for which the function does not change sign. Courant's nodal domain theorem gives a bound on the number of nodal domains of eigenfunctions of elliptic operators. In particular, the k-th eigenfunction contains no more tha...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 539; pp. 60 - 71
Main Author Urschel, John C.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 15.02.2018
American Elsevier Company, Inc
Subjects
Decomposition
Eigenvalues
Eigenvectors
Generalized Laplacian
Graphs
Laplace transforms
Linear algebra
Mathematical functions
Nodal domains
Theorems
Vector space
05C50
Nodal domains
Generalized Laplacian
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Summary:A nodal domain of a function is a maximally connected subset of the domain for which the function does not change sign. Courant's nodal domain theorem gives a bound on the number of nodal domains of eigenfunctions of elliptic operators. In particular, the k-th eigenfunction contains no more than k nodal domains. We prove a generalization of Courant's theorem to discrete graphs. Namely, we show that for the k-th eigenvalue of a generalized Laplacian of a discrete graph, there exists a set of corresponding eigenvectors such that each eigenvector can be decomposed into at most k nodal domains. In addition, we show this set to be of co-dimension zero with respect to the entire eigenspace.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2017.11.003