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On the Maximum Spread of Non-Negative Matrices

Susie Lu, John Urschel

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.

On the Maximum Spread of Non-Negative Matrices

Abstract

Given a directed graph , the spread of is the largest distance between any two eigenvalues of its adjacency matrix. In 2022, Breen, Riasanovsky, Tait, and Urschel asked what -vertex directed graph maximizes spread, and whether this graph is undirected. We prove the more general result that the spread of any non-negative matrix with is at most , which is tight up to an additive factor and exact when is a multiple of three. Furthermore, our results show that the matrix with maximum spread is always symmetric.

Paper Structure

This paper contains 1 section, 8 theorems, 29 equations, 1 figure.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

Let $A$ be an $n \times n$ symmetric non-negative matrix. Then with equality if and only if $n = 0 \bmod 3$ and $A = P_{\pi} \left(\left(\right) \otimes J_{\frac{n}{3}}\right)P_{\pi}$, where $J_k$ is the $k \times k$ all-ones matrix and $P_{\pi}$ is a permutation matrix. Furthermore:

Figures (1)

  • Figure 1: (a) The left plot shows $y = f(\frac{\lambda_{\max}}{n},0)$ (blue) and $y = \frac{21}{16}$ (orange). The blue curve is below the orange line for all $x \not\in [0.85177, 0.89726]$, which is marked by red dashed lines. (b) The right plot shows $y = f(\frac{\lambda_{\max}}{n},\frac{1}{200})$ (blue), which is always below $y = \frac{21}{16}$ (orange). A simple calculation shows that the maximum value of $f(\frac{\lambda_{\max}}{n},\frac{1}{200})$ is approximately 1.31229, which is less than $\frac{21}{16}$.

Theorems & Definitions (12)

  • Theorem 1: Breen22
  • Theorem 2: Directed Spread Theorem
  • Lemma 3
  • proof : Proof of Lemma \ref{['lm:bounds']}
  • Corollary 4
  • proof
  • Lemma 5: Bendixson's inequality, Bendixson
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 2 more