Nodal domain count for the generalized graph $p$-Laplacian

TL;DR

This paper investigates the spectral properties of a generalized $p$-Laplacian on graphs, focusing on nodal domain counts, and extends classical inequalities to this nonlinear setting, revealing new insights especially for the linear case.

Contribution

It introduces new bounds on nodal domains for the generalized $p$-Laplacian and extends Weyl's inequalities to the nonlinear graph operator setting.

Findings

Spectral properties characterized for the generalized $p$-Laplacian.

Upper and lower bounds on nodal domains established.

Extension of Weyl's inequalities to nonlinear operators.

Abstract

Inspired by the linear Schr\"odinger operator, we consider a generalized $p$-Laplacian operator on discrete graphs and present new results that characterize several spectral properties of this operator with particular attention to the nodal domain count of its eigenfunctions. Just like the one-dimensional continuous $p$-Laplacian, we prove that the variational spectrum of the discrete generalized $p$-Laplacian on forests is the entire spectrum. Moreover, we show how to transfer Weyl's inequalities for the Laplacian operator to the nonlinear case and prove new upper and lower bounds on the number of nodal domains of every eigenfunction of the generalized $p$-Laplacian on generic graphs, including variational eigenpairs. In particular, when applied to the linear case $p=2$, in addition to recovering well-known features, the new results provide novel properties of the linear Schr\"odinger operator.