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 lambda(2) 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 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 Verdieres upper bound on the maximal lambda(2)-multiplicity, m <= chr(gamma(G))-1, where chr(gamma(G)) is the chromatic number and gamma(G) is the genus of G.