A Bose-Laskar-Hoffman theory for $μ$-bounded graphs with fixed smallest eigenvalue
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909972839792640 |
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| author | Koolen, Jack H. Ge, Hong-Jun Lv, Chenhui Yang, Qianqian |
| author_facet | Koolen, Jack H. Ge, Hong-Jun Lv, Chenhui Yang, Qianqian |
| contents | In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and instead employ a novel approach that combines the Bose-Laskar type argument with Hoffman theory to derive structural insights into $μ$-bounded graphs with fixed smallest eigenvalue. Our method establishes a reasonable bound on the minimum degree. Note that local graphs of distance-regular graphs are $μ$-bounded. We apply these results to characterize the structure for any local graph of a distance-regular graph with classical parameters $(D,b,α,β)$. Consequently, we show that the parameter $α$ is bounded by a cubic polynomial in $b$ if $D \geq 9$ and $b \geq 2$. We also show that $α\leq 2$ if $b =2$ and $D \geq 12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Bose-Laskar-Hoffman theory for $μ$-bounded graphs with fixed smallest eigenvalue Koolen, Jack H. Ge, Hong-Jun Lv, Chenhui Yang, Qianqian Combinatorics 05C50, 05C75, 05E30 In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and instead employ a novel approach that combines the Bose-Laskar type argument with Hoffman theory to derive structural insights into $μ$-bounded graphs with fixed smallest eigenvalue. Our method establishes a reasonable bound on the minimum degree. Note that local graphs of distance-regular graphs are $μ$-bounded. We apply these results to characterize the structure for any local graph of a distance-regular graph with classical parameters $(D,b,α,β)$. Consequently, we show that the parameter $α$ is bounded by a cubic polynomial in $b$ if $D \geq 9$ and $b \geq 2$. We also show that $α\leq 2$ if $b =2$ and $D \geq 12$. |
| title | A Bose-Laskar-Hoffman theory for $μ$-bounded graphs with fixed smallest eigenvalue |
| topic | Combinatorics 05C50, 05C75, 05E30 |
| url | https://arxiv.org/abs/2502.05520 |