On the maximum spread of non-negative matrices

Abstract

Given a directed graph G, the spread of G is the largest distance between any two eigenvalues of its adjacency matrix. In 2022, Breen, Riasanovsky, Tait, and Urschel asked what n-vertex directed graph maximizes spread, and whether this graph is undirected. We prove the more general result that the spread of any $n\times n$ non-negative matrix A with ${\Vert A\Vert }_{\max }\le 1$ is at most $2n/\sqrt{3}$, which is tight up to an additive factor and exact when n is a multiple of three. Furthermore, our results show that the matrix with maximum spread is always symmetric.