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.

Mathematics Subject Classification: 05C50 Graphs and matrices, 05C22 Signed, gain and biased graphs, 05C83 Graph minors

Key Words: graph theory, graph Laplacian, eigenvectors, nodal domain theorems,bipartite graphs

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