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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

This note discusses a relation between the multiplicity m of the second eigenvalue λ2 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 λ2 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 Verdière's upper bound on the maximal λ2-multiplicity, where chr(γ(G)) is the chromatic number and γ(G) is the genus of G.