On the chromatic number of a simplicial complex

In [Ho] A.J. Hoffman proved a lower bound on the chromatic number of a graph in the terms of the largest and the smallest eigenvalues of its adjacency matrix. In this paper, we prove a higher dimensional version of this result and give a lower bound on the chromatic number of a pure $d$-dimensional simplicial complex in the terms of the spectra of the higher Laplacian operators.