对称矩阵、符号图和节域定理
Symmetric Matrices, Signed Graphs, and Nodal Domain Theorems
摘要：
在2001年，Davies，Gladwell，Leydold和Stadler证明了广义Laplacian特征函数的离散节点域定理，即，非正对角元素的对称矩阵。本文利用导出符号图的结构，建立了任意对称矩阵的节域定理。我们的概念的节点域的任何功能在一个有符号的图是开关不变的。当导出符号图是平衡的时，我们的定义和上界估计退化为广义拉普拉斯算子的已有结果。我们的方法提供了一个更概念性的理解Fiedler的结果的特征函数的非循环矩阵。这个新的观点导致强节点域个数的下界估计，改进了Berkolaiko和Xu-Yau的结果。我们还证明了一种新的类型的下界估计的对偶论证。
英文摘要：
In 2001, Davies, Gladwell, Leydold, and Stadler proved discrete nodal domain theorems for eigenfunctions of generalized Laplacians, i.e., symmetric matrices with non-positive off-diagonal entries. In this paper, we establish nodal domain theorems for arbitrary symmetric matrices by exploring the induced signed graph structure. Our concepts of nodal domains for any function on a signed graph are switching invariant. When the induced signed graph is balanced, our definitions and upper bound estimates reduce to existing results for generalized Laplacians. Our approach provides a more conceptual understanding of Fiedler's results on eigenfunctions of acyclic matrices. This new viewpoint leads to lower bound estimates for the number of strong nodal domains which improves previous results of Berkolaiko and Xu-Yau. We also prove a new type of lower bound estimates by a duality argument.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
