Nodal Domain Theorems and Bipartite Subgraphs
Türker Biyikoglu, Josef Leydold, and Peter F. Stadler
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Submission date: 02. Aug. 2005 (revised version: August 2005)
published in: The electronic journal of linear algebra, 13 (2005), p. 344-351
DOI number (of the published article): 10.13001/1081-3810.1167
MSC-Numbers: 05C50, 05C22, 05C83
Keywords and phrases: graph laplacian, nodal domain theorem, eigenvectors, bipartite graphs
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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.