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Some properties of Laplacian eigenvectors

Gutman Ivan (Faculty of Science University of Kragujevac)

Let G be a graph on n vertices, G its complement and Kn the complete graph on n vertices. We show that if G is connected, then any Laplacian eigenvector of G is also a Laplacian eigenvector of G and of Kn . This result holds, with a slight modification, also for disconnected graphs. We establish also some other results, all showing that the structural information contained in the Laplacian eigenvectors is rather limited. An analogy between the theories of Laplacian and ordinary graph spectra is pointed out.

Keywords: Laplacian spectrum, Laplacian matrix, Laplacian eigenvector (of graph), Laplacian eigenvalue (of graph)