论文信息 - Nodal Domain Theorems and Bipartite Subgraphs

Nodal Domain Theorems and Bipartite Subgraphs

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. The number of strong nodal domains is shown not to 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.

[1] J. Leydold,et al. Discrete Nodal Domain Theorems , 2000, math/0009120.

[2] Peter F. Stadler,et al. Graph Laplacians, nodal domains, and hyperplane arrangements , 2004 .

[3] R. Roth,et al. On the eigenvectors belonging to the minimum eigenvalue of an essentially nonnegative symmetric matrix with bipartite graph , 1989 .

[4] Dr. M. G. Worster. Methods of Mathematical Physics , 1947, Nature.

[5] David S. Johnson,et al. Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[6] B. Mohar. Some applications of Laplace eigenvalues of graphs , 1997 .

[7] Josef Leydold,et al. Faber-Krahn type inequalities for trees , 2007, J. Comb. Theory, Ser. B.

[8] Türker Bıyıkoğlu,et al. A discrete nodal domain theorem for trees , 2003 .

[9] M. Fiedler. Eigenvectors of acyclic matrices , 1975 .

[10] J. Friedman. Some geometric aspects of graphs and their eigenfunctions , 1993 .

[11] I. Chavel. Eigenvalues in Riemannian geometry , 1984 .