  We apply Cauchy's interlacing theorem to derive some eigenvalue bounds to the
chromatic number using the normalized Laplacian matrix, including a
combinatorial characterization of when equality occurs. Further, we introduce
some new expansion type of parameters which generalize the Cheeger constant of
a graph, and relate them to the colourings which meet our eigenvalue bound with
equality. Finally, we exhibit a family of examples, which include the graphs
that appear in the statement of the Erd\H{o}s-Faber-Lov\'asz conjecture.
