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

In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with com-ponents equal to {-1, 0, 1}. We extend the graph parameter max k-cut to square matrices and prove a general sharp up-per bound, which implies upper bounds on the max k-cut of a graph using the smallest signless Laplacian eigenvalue, the smallest adjacency eigenvalue, and the largest Laplacian eigenvalue of the graph. In addition, we construct infinite fam-ilies of extremal graphs for the obtained upper bounds. (c) 2023 Elsevier Inc. All rights reserved.