Nodal Domains of Eigenvectors for $1$-Laplacian on Graphs

K. C. Chang; Sihong Shao; Dong Zhang

doi:10.1016/j.aim.2016.12.020

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arXiv · 1602.07472

Nodal Domains of Eigenvectors for $1$-Laplacian on Graphs

K. C. Chang·Sihong Shao·Dong Zhang

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Abstract

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 in [{\sl K.~C. Chang, Spectrum of the $1$-Laplacian and Cheeger constant on graphs, J. Graph Theor., DOI: 10.1002/jgt.21871}], to confirm that the critical values obtained by the minimax principle may not cover all eigenvalues of graph $1$-Laplacian.

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BibTeX RIS

K. C. Chang, Sihong Shao, Dong Zhang. 2016-02-24. Nodal Domains of Eigenvectors for $1$-Laplacian on Graphs. https://doi.org/10.1016/j.aim.2016.12.020

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