Digraphs with few distinct eigenvalues

Abstract

This paper provides insight into the problem of characterizing digraphs (with loops permitted) that have few distinct adjacency eigenvalues, or equivalently, characterizing square $(0,1)$-matrices that have few distinct eigenvalues. A spectral characterization of strongly connected digraphs whose adjacency matrix has exactly two distinct eigenvalues is given and constructions of such digraphs are described. In addition, bipartite digraphs with exactly three distinct eigenvalues are discussed.