Bounding the parameter $β$ of a distance-regular graph with classical parameters
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| Format: | Preprint |
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2024
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| _version_ | 1866918255793274880 |
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| author | Lv, Chenhui Koolen, Jack H. |
| author_facet | Lv, Chenhui Koolen, Jack H. |
| contents | Let $Γ$ be a distance-regular graph with classical parameters $(D, b, α, β)$ satisfying $b\geq 2$ and $D\geq 3$. Let $r=1+b+b^2+\cdots+b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(α,b)$ only depending on $α$ and $b$, such that if $β\geq C(α, b)r^2$, then either $Γ$ is a Grassmann graph or a bilinear forms graph.
In this work, we show that for $b\geq 2$ and $D\geq 3$, then there exists a constant $C_1(α, b)$ only depending on $α$ and $b$, such that if $β\geq C_1(α, b)r$, then either $Γ$ is a Grassmann graph, or a bilinear forms graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_22994 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounding the parameter $β$ of a distance-regular graph with classical parameters Lv, Chenhui Koolen, Jack H. Combinatorics 05E30 Let $Γ$ be a distance-regular graph with classical parameters $(D, b, α, β)$ satisfying $b\geq 2$ and $D\geq 3$. Let $r=1+b+b^2+\cdots+b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(α,b)$ only depending on $α$ and $b$, such that if $β\geq C(α, b)r^2$, then either $Γ$ is a Grassmann graph or a bilinear forms graph. In this work, we show that for $b\geq 2$ and $D\geq 3$, then there exists a constant $C_1(α, b)$ only depending on $α$ and $b$, such that if $β\geq C_1(α, b)r$, then either $Γ$ is a Grassmann graph, or a bilinear forms graph. |
| title | Bounding the parameter $β$ of a distance-regular graph with classical parameters |
| topic | Combinatorics 05E30 |
| url | https://arxiv.org/abs/2410.22994 |