Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques naturelles / sciences mathematiques Vol. CXXVII, No. 28, pp. 1–6 (2003) 

Some properties of Laplacian eigenvectorsI. GutmanAbstract: Let $G$ be a graph on $n$ vertices, $\bar G$ its complement and $K_n$ 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 $\bar G$ and of $K_n$ . 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) Classification (MSC2000): 05C50 Full text of the article: (for faster download, first choose a mirror)
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