On the Maximum Spread of Non-Negative Matrices

TL;DR

This paper establishes an upper bound on the spread of non-negative matrices and shows that the maximum spread is achieved by symmetric matrices, answering a question about directed graphs and their adjacency matrices.

Contribution

It proves a tight upper bound on the spread of non-negative matrices and demonstrates that the maximum spread matrix must be symmetric.

Findings

Maximum spread of non-negative matrices is at most 2n/√3.

The maximum spread matrix is always symmetric.

Bound is tight up to an additive factor for multiples of three.

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 $\|A\|_{\max} \le 1$ 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.