Symmetric matrices, signed graphs, and nodal domain theorems

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. this paper, we establish arbitrary by exploring the induced signed graph structure. Our concepts domains any function on a are switching invariant. When is balanced, our definitions upper bound estimates reduce to existing results Laplacians. approach provides more conceptual understanding Fiedler’s acyclic matrices. This new viewpoint leads lower number strong which improves previous Berkolaiko Xu–Yau. We also prove type duality argument.