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

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Some properties of Laplacian eigenvectors

I. Gutman

Abstract: 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

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Electronic fulltext finalized on: 17 Sep 2003. This page was last modified: 20 Jun 2011.

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