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 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 is discussed. For a certain class of graphs, it is shown 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 Verdiére’s upper bound on the maximal λ2-multiplicity, m ≤ chr(γ(G))−1, where chr(γ(G)) is the chromatic number and γ(G) is the genus of G.