Nodal domain theorems for -Laplacians on signed graphs
Chuanyuan Ge
University of Science and Technology of China, Hefei, ChinaShiping Liu
University of Science and Technology of China, Hefei, ChinaDong Zhang
Peking University, Beijing, China
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Abstract
We establish various nodal domain theorems for -Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph -Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we obtain a higher order Cheeger inequality that relates the variational eigenvalues of -Laplacians and Atay–Liu's multi-way Cheeger constants on signed graphs. In the particular case of , this leads to several identities relating variational eigenvalues and multi-way Cheeger constants. Intriguingly, our approach also leads to new results on usual graphs, including a weak version of Sturm's oscillation theorem for graph -Laplacians and nonexistence of eigenvalues between the largest and second largest variational eigenvalues of -Laplacians with on connected bipartite graphs.
Cite this article
Chuanyuan Ge, Shiping Liu, Dong Zhang, Nodal domain theorems for -Laplacians on signed graphs. J. Spectr. Theory 13 (2023), no. 3, pp. 937–989
DOI 10.4171/JST/472