标题： 节点特征向量在图上的$1$-拉普拉斯的节点域 显示英文标题

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

摘要： 图$1$-Laplacian 的特征向量具有一些局域化性质：一方面，特征向量的节点域仍然是具有相同特征值的特征向量；另一方面，可以通过一些特殊技巧，将几个基本特征分量和模块组合起来，形成新图的特征向量，这些分量和模块具有相同的特征值。 图的 Courant 节点域定理被扩展到图$1$-Laplacian 的强节点域，但对于弱节点域来说，该定理不成立。 为了对独立特征向量的数量提供更精确的估计，引入了代数重数的概念。 对 [{\sl K.~C. Chang,$1$-Laplacian 的谱和图上的 Cheeger 常数, J. Graph Theor., DOI: 10.1002/jgt.21871}] 中提出的问题给出了肯定回答，以确认通过极值原理得到的临界值可能无法覆盖图$1$-Laplacian 的所有特征值。 显示英文摘要

摘要： 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.