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| Hauptverfasser: | , , , , , , , |
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| Format: | Preprint |
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2023
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| Online-Zugang: | https://arxiv.org/abs/2311.04689 |
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| _version_ | 1866912000394657792 |
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| author | Bouthat, Ludovick Chávez, Ángel Fullerton, Sarah LaFortune, Matilda Linarez, Keyron Liyanage, Nethmin Son, Justin Ting, Tyler |
| author_facet | Bouthat, Ludovick Chávez, Ángel Fullerton, Sarah LaFortune, Matilda Linarez, Keyron Liyanage, Nethmin Son, Justin Ting, Tyler |
| contents | Recent work shows that a new family of norms on Hermitian matrices arise by evaluating the even degree complete homogeneous symmetric (CHS) polynomials on the eigenvalues of a Hermitian matrix. The CHS norm of a graph is then defined by evaluating the even degree CHS polynomials on the eigenvalues of the adjacency matrix of a graph. The fact that these norms are defined in terms of eigenvalues (as opposed to singular values) ensures they can distinguish between graphs that other norms cannot. In addition, we prove that the CHS norms are minimized over all connected graphs by the path and maximized over all connected graphs by the complete graph. Finally, we prove that the CHS norms are minimized over all trees by the path and maximized over all trees by the star. Our paper is intended for a wide mathematical audience and we assume no prior knowledge about graphs or symmetric polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04689 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extremal Polynomial Norms of Graphs Bouthat, Ludovick Chávez, Ángel Fullerton, Sarah LaFortune, Matilda Linarez, Keyron Liyanage, Nethmin Son, Justin Ting, Tyler Combinatorics Recent work shows that a new family of norms on Hermitian matrices arise by evaluating the even degree complete homogeneous symmetric (CHS) polynomials on the eigenvalues of a Hermitian matrix. The CHS norm of a graph is then defined by evaluating the even degree CHS polynomials on the eigenvalues of the adjacency matrix of a graph. The fact that these norms are defined in terms of eigenvalues (as opposed to singular values) ensures they can distinguish between graphs that other norms cannot. In addition, we prove that the CHS norms are minimized over all connected graphs by the path and maximized over all connected graphs by the complete graph. Finally, we prove that the CHS norms are minimized over all trees by the path and maximized over all trees by the star. Our paper is intended for a wide mathematical audience and we assume no prior knowledge about graphs or symmetric polynomials. |
| title | Extremal Polynomial Norms of Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.04689 |