  We consider several semidefinite programming relaxations for the max-$k$-cut
problem, with increasing complexity. The optimal solution of the weakest
presented semidefinite programming relaxation has a closed form expression that
includes the largest Laplacian eigenvalue of the graph under consideration.
This is the first known eigenvalue bound for the max-$k$-cut when $k>2$ that is
applicable to any graph. This bound is exploited to derive a new eigenvalue
bound on the chromatic number of a graph. For regular graphs, the new bound on
the chromatic number is the same as the well-known Hoffman bound; however, the
two bounds are incomparable in general. We prove that the eigenvalue bound for
the max-$k$-cut is tight for several classes of graphs. We investigate the
presented bounds for specific classes of graphs, such as walk-regular graphs,
strongly regular graphs, and graphs from the Hamming association scheme.
