Nodal domain theorems for $p$-Laplacians on signed graphs

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 D 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.