Symmetric matrices, signed graphs, and nodal domain theorems

被引:3
|
作者
Ge, Chuanyuan [1 ]
Liu, Shiping [1 ]
机构
[1] Univ Sci & Technol China, Sch Math Sci, Hefei 230026, Peoples R China
基金
中国国家自然科学基金;
关键词
39A12; 05C22; 05C50; 15A18; PERRON-FROBENIUS; EIGENVECTORS; DISCRETE; EIGENVALUE; PROPERTY;
D O I
10.1007/s00526-023-02479-6
中图分类号
O29 [应用数学];
学科分类号
070104 ;
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.
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页数:30
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