Signed graphs with exactly two main eigenvalues: The unicyclic case
Fuente:
arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866912942613594112 |
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| author | Du, Zenan Liu, Fenjin Liu, Hechao Lin, Jifu Yang, Wenxu |
| author_facet | Du, Zenan Liu, Fenjin Liu, Hechao Lin, Jifu Yang, Wenxu |
| contents | An eigenvalue $λ$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Signed graphs with exactly two main eigenvalues: The unicyclic case Du, Zenan Liu, Fenjin Liu, Hechao Lin, Jifu Yang, Wenxu Combinatorics An eigenvalue $λ$ of a signed graph $S$ of order $n$ is called a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector $j$. Characterizing signed graphs with exactly $k$ $(1\le k\le n)$ distinct main eigenvalues is a problem in algebraic and graph theory that has been studied since 2020. Du et al. (2024, 2026) characterized a class of signed graphs with exactly two main eigenvalues by analyzing a type of multigraph whose base graph is a tree. In this paper, we extend this study to the case where the associated multigraph has a unicyclic base graph, and we conclude by proposing several open problems. |
| title | Signed graphs with exactly two main eigenvalues: The unicyclic case |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.04063 |