A structure theory for signed graphs with fixed smallest eigenvalue
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866917291417927680 |
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| author | Koolen, Jack H. Liu, Jing-Yuan Yang, Qianqian Cao, Meng-Yue |
| author_facet | Koolen, Jack H. Liu, Jing-Yuan Yang, Qianqian Cao, Meng-Yue |
| contents | In this paper, we will give a structure theory for signed graphs with fixed smallest eigenvalue and investigate signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. Given a real number $λ\leq -1$, we show that the following hold for each signed graph $(G,σ)$ with smallest eigenvalue at least $λ$ and large minimum valency: $\mathrm{(i)}$ there exist dense induced subgraphs $N_1, \dots, N_r$ in $(G,σ)$ such that each vertex lies in at most $\lfloor -λ\rfloor$ $N_i$'s and almost all edges of $(G,σ)$ lie in at least one of the $N_i$'s; $\mathrm{(ii)}$ if $λ>-1-\sqrt{2}$, then $(G,σ)$ has smallest eigenvalue at least $-2$ and $(G,σ)$ is $1$-integrable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20783 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A structure theory for signed graphs with fixed smallest eigenvalue Koolen, Jack H. Liu, Jing-Yuan Yang, Qianqian Cao, Meng-Yue Combinatorics 05C50, 05C22, 52C35, 05C62, 05C75 In this paper, we will give a structure theory for signed graphs with fixed smallest eigenvalue and investigate signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. Given a real number $λ\leq -1$, we show that the following hold for each signed graph $(G,σ)$ with smallest eigenvalue at least $λ$ and large minimum valency: $\mathrm{(i)}$ there exist dense induced subgraphs $N_1, \dots, N_r$ in $(G,σ)$ such that each vertex lies in at most $\lfloor -λ\rfloor$ $N_i$'s and almost all edges of $(G,σ)$ lie in at least one of the $N_i$'s; $\mathrm{(ii)}$ if $λ>-1-\sqrt{2}$, then $(G,σ)$ has smallest eigenvalue at least $-2$ and $(G,σ)$ is $1$-integrable. |
| title | A structure theory for signed graphs with fixed smallest eigenvalue |
| topic | Combinatorics 05C50, 05C22, 52C35, 05C62, 05C75 |
| url | https://arxiv.org/abs/2602.20783 |