  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.
