A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem and the substantial dimension of tight polyhedral surfaces

Details A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem and the substantial dimension of tight polyhedral surfaces A relation between the multiplicity of the second eigenvalue of a graph Laplacian, Courant's nodal line theorem and the substantial dimension of tight polyhedral surfaces

2010-07-23

arxiv.org/abs/1007.4132v1

ELA, Volume 16, pp. 315-324, October 2007

Physics

Mathematical Physics

Keywords: Graph Laplacian; Tight Embedding; Nodal domain; Eigenfunctions; Polyhedral Manifolds. http://www.math.technion.ac.

Scientific paper

This note discusses a relation between the multiplicity m of the second eigenvalue {\lambda}2 of a Laplacian on a graph G, tight mappings of G and a discrete analogue of Courant's nodal line theorem. For a certain class of graphs, we show that the m-dimensional eigenspace of {\lambda}2 is tight and thus defines a tight mapping of G into an m-dimensional Euclidean space. The tightness of the mapping is shown to set Colin de Verdi\`ere's upper bound on the maximal {\lambda}2-multiplicity, where chr({\gamma}(G)) is the chromatic number and {\gamma}(G) is the genus of G.

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