%0 Journal Article %J Advances in Mathematics %D 2017 %T Nodal domains of eigenvectors for 1-Laplacian on graphs %A K.-C. Chang %A Sihong Shao %A Dong Zhang %X The eigenvectors for graph 1-Laplacian possess some sort of localization property: On one hand, the characteristic function on 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 Chang (2016) [3], to confirm that the critical values obtained by the minimax principle may not cover all eigenvalues of graph 1-Laplacian. %B Advances in Mathematics %V 308 %P 529-574 %G eng %U http://dx.doi.org/10.1016/j.aim.2016.12.020