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

TL;DR

This paper develops nodal domain theorems for $p$-Laplacians on signed graphs, unifying existing results and deriving new inequalities and properties, including a higher order Cheeger inequality and results specific to $p=1$ and bipartite graphs.

Contribution

It introduces unified nodal domain theorems for $p$-Laplacians on signed graphs, extending classical results and establishing new inequalities and spectral properties.

Findings

Unified nodal domain theorems for $p$-Laplacians on signed graphs

Higher order Cheeger inequality relating eigenvalues and Cheeger constants

Weak Sturm's oscillation theorem and eigenvalue nonexistence results for specific graph classes

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.