论文标题
对称矩阵，签名图和节点域定理
Symmetric Matrices, Signed Graphs, and Nodal Domain Theorems
论文作者
论文摘要
2001年，戴维斯（Davies），格拉德威尔（Gladwell），莱多德（Leydold）和斯塔德勒（Stadler）证明了对广义拉普拉斯人（即具有非阳性异性词的对称矩阵）的特征性特征性的离散结构域定理。在本文中，我们通过探索诱导的符号图结构来建立任意对称矩阵的节点域定理。我们对签名图上任何功能的节点域的概念都在切换不变。当诱导的签名图平衡时，我们的定义和上限估计值将普遍的拉普拉斯人的现有结果减少。我们的方法为Fiedler对无环矩阵的本征函数的结果提供了更概念性的理解。这种新的观点导致对强节结构域数量的较低估计，从而改善了Berkolaiko和Xu-Yau的先前结果。我们还通过二元参数证明了一种新型的下限估计。
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.
