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

TL;DR

This paper investigates the properties of eigenvectors of the graph 1-Laplacian, including localization, nodal domain behavior, and eigenvalue multiplicities, extending classical theorems and addressing open questions.

Contribution

It extends the Courant nodal domain theorem to strong nodal domains for the graph 1-Laplacian, introduces algebraic multiplicity for eigenvalues, and clarifies the relationship between minimax critical values and all eigenvalues.

Findings

Nodal domains of eigenvectors exhibit localization properties.

The Courant nodal domain theorem is extended to strong nodal domains.

Critical values from minimax principles may not encompass all eigenvalues.

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.