On the Maximum Spread of Directed Graphs

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 × n non-negative matrix A with \|A\|ₘₐₓ ≤ 1 is at most 2n/√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.