On graphs with adjacency and signless Laplacian matrices eigenvectors entries in {−1,+1}

Abstract In 1986, Herbert Wilf asked what kind of graphs have an eigenvector with entries formed only by ±1. In this paper, we answer this question for the adjacency, Laplacian, and signless Laplacian matrices of a graph. To this end, we generalize the concept of an exact graph to the adjacency and signless Laplacian matrices. Infinite families of exact graphs for all those matrices are presented.