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In this paper, we characterize all graphs with eigenvectors of the signlessLaplacian and adjacency matrices with components equal to $\{- 1, 0, 1\}.$ Weextend the graph parameter max $k$-cut to square matrices and prove a generalsharp upper bound, which implies upper bounds on the max $k$-cut of a graphusing the smallest signless Laplacian eigenvalue, the smallest adjacencyeigenvalue, and the largest Laplacian eigenvalue of the graph. In addition, weconstruct infinite families of extremal graphs for the obtained upper bounds.

On graphs with eigenvectors in $\{1, -1, 0\}$ and the max $k$-cut problem