Nodal domain theorem of signed hypergraphs

TL;DR

This paper extends the nodal domain theorem to signed hypergraphs, providing theoretical insights into eigenfunctions of their Laplacians and establishing bounds on the number of strong nodal domains.

Contribution

It introduces a nodal domain theorem for the normalized Laplace operator in signed hypergraphs, a novel generalization from previous graph-based results.

Findings

Established nodal domain theorems for signed hypergraphs.

Provided lower bounds for the number of strong nodal domains.

Abstract

In 2001, Davies, Gladwell, Leydold, and Stadler proved discrete nodal domain theorems for eigenfunctions of generalized Laplacians. In 2019, Jost and Mulas generalized the normalized combinatorial Laplace operator of graphs to signed hypergraphs. In this paper, we establish nodal domain theorems for the normalized combinatorial Laplace operator in signed hypergraphs. We also obtain a lower bound estimates for the number of strong nodal domains.