Delve into the future of research at MiS with our preprint repository. Our scientists are making groundbreaking discoveries and sharing their latest findings before they are published. Explore repository to stay up-to-date on the newest developments and breakthroughs.
Nodal Domain Theorems and Bipartite Subgraphs
Türker Biyikoglu, Josef Leydold and Peter F. Stadler
The Discrete Nodal Domain Theorem states that an eigenfunction of the $k$-th largest eigenvalue of a generalized graph Laplacian has at most $k$ (weak) nodal domains. We show that the number of strong nodal domains cannot exceed the size of a maximal induced bipartite subgraph and that this bound is sharp for generalized graph Laplacians. Similarly, the number of weak nodal domains is bounded by the size of a maximal bipartite minor.