Abstract
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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Acknowledgements
This work has been presented in the conference, Mathematical concepts in the Science and Humanities (MPI MiS, May 16-25,2022), which is dedicated to Professor Jürgen Jost on the occasion of his 65th birthday. We are very grateful to Dong Zhang for discussions on discrete nodal domain theorems of signless Laplacians. We thank Ali Mohammadian for bringing his interesting work [32] to our attention after the submission of our first arXiv version. This work is supported by the National Key R and D Program of China 2020YFA0713100 and the National Natural Science Foundation of China (No. 12031017).
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Communicated by Andrea Mondino.
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