Symmetric Matrices, Signed Graphs, and Nodal Domain Theorems

TL;DR

This paper extends nodal domain theorems from generalized Laplacians to all symmetric matrices by analyzing the associated signed graph structures, providing new bounds and conceptual insights.

Contribution

It introduces a unified framework for nodal domain theorems applicable to any symmetric matrix via signed graph analysis, improving existing bounds and understanding.

Findings

Established nodal domain theorems for arbitrary symmetric matrices.

Provided improved lower bounds for the number of strong nodal domains.

Developed a duality-based approach for new lower bound estimates.

Abstract

In 2001, Davies, Gladwell, Leydold, and Stadler proved discrete nodal domain theorems for eigenfunctions of generalized Laplacians, i.e., symmetric matrices with non-positive off-diagonal entries. In this paper, we establish nodal domain theorems for arbitrary symmetric matrices by exploring the induced signed graph structure. Our concepts of nodal domains for any function on a signed graph are switching invariant. When the induced signed graph is balanced, our definitions and upper bound estimates reduce to existing results for generalized Laplacians. Our approach provides a more conceptual understanding of Fiedler's results on eigenfunctions of acyclic matrices. This new viewpoint leads to lower bound estimates for the number of strong nodal domains which improves previous results of Berkolaiko and Xu-Yau. We also prove a new type of lower bound estimates by a duality argument.