On graphs with adjacency and signless Laplacian matrix eigenvectors entries in $\{-1, +1\}$

TL;DR

This paper characterizes graphs whose eigenvectors of adjacency, Laplacian, and signless Laplacian matrices have entries only in  , answering a question posed in 1986 and introducing the concept of exact graphs for these matrices.

Contribution

It provides a complete characterization of graphs with   eigenvectors and generalizes the concept of exact graphs to adjacency and signless Laplacian matrices.

Findings

Identifies graphs with   eigenvectors for key matrices

Introduces the concept of exact graphs for adjacency and signless Laplacian matrices

Constructs infinite families of such graphs

Abstract

Let $G$ be a simple graph. In 1986, Herbert Wilf asked what kind of graphs have an eigenvector with entries formed only by $\pm 1$? In this paper, we answer this question for the adjacency, Laplacian and signless Laplacian matrix of a graph. Besides, we generalize the concept of an exact graph to the adjacency and signless Laplacian matrices. Infinity families of exact graphs for all those matrices are presented.