<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">K.-C. Chang</style></author><author><style face="normal" font="default" size="100%">Sihong Shao</style></author><author><style face="normal" font="default" size="100%">Dong Zhang</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Nodal domains of eigenvectors for 1-Laplacian on graphs</style></title><secondary-title><style face="normal" font="default" size="100%">Advances in Mathematics</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2017</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://dx.doi.org/10.1016/j.aim.2016.12.020</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">308</style></volume><pages><style face="normal" font="default" size="100%">529-574</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">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.</style></abstract></record></records></xml>