Abstract
We establish various nodal domain theorems for p -Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph p -Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we obtain a higher order Cheeger inequality that relates the variational eigenvalues of p -Laplacians and Atay–Liu's multi-way Cheeger constants on signed graphs. In the particular case of p=1 , 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 1 -Laplacians and nonexistence of eigenvalues between the largest and second largest variational eigenvalues of p -Laplacians with p>1 on connected bipartite graphs.
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