There exist inequalities relating the maximum and minimum eigenvalues of the adjacency matrix of a graph with its chromatic numbers, i.e. the Wilf's and Hoffmann's inequalities, which put together state that $1-\frac{\lambda_{max}}{\lambda_{min}}\le\chi(G)\le 1+\lambda_{max}$, where $\chi, \lambda_{max}, \lambda_{min}$ respectively stand for the chromatic number, maximum and minimum eigenvalues of the adjacency matrix of $G$. But, for regular graphs, the upper bound by Wilf's inequality is quite trivial, that is same as the greedy coloring bound.

Is there a better bound for the chromatic number of a regular graphs using the spectra of, say the adjacency or Laplacian matrices? Thanks beforehand.