On the Maximum Spread of Non-Negative Matrices
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917082687340544 |
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| author | Lu, Susie Urschel, John |
| author_facet | Lu, Susie Urschel, John |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05760 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Maximum Spread of Non-Negative Matrices Lu, Susie Urschel, John Combinatorics 05C50 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. |
| title | On the Maximum Spread of Non-Negative Matrices |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2508.05760 |