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| Main Authors: | , , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2204.07842 |
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| _version_ | 1866917662238441472 |
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| author | Xia, Zheng-Jiang Lee, Jae-Ho Koolen, Jack H. |
| author_facet | Xia, Zheng-Jiang Lee, Jae-Ho Koolen, Jack H. |
| contents | Let $Γ$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $θ_0>θ_1>θ_2>θ_3>θ_4$. Then $Γ$ is tight in the sense of Jurišić, Koolen, and Terwilliger [12] whenever $Γ$ is locally strongly regular with nontrivial eigenvalues $p:=θ_2$ and $-q:=θ_3$. Assume that $Γ$ is tight. Then the intersection numbers of $Γ$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $Γ$. We denote $Γ$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph. Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_07842 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A New Feasibility Condition for the AT4 Family Xia, Zheng-Jiang Lee, Jae-Ho Koolen, Jack H. Combinatorics 05E30 Let $Γ$ be an antipodal distance-regular graph with diameter $4$ and eigenvalues $θ_0>θ_1>θ_2>θ_3>θ_4$. Then $Γ$ is tight in the sense of Jurišić, Koolen, and Terwilliger [12] whenever $Γ$ is locally strongly regular with nontrivial eigenvalues $p:=θ_2$ and $-q:=θ_3$. Assume that $Γ$ is tight. Then the intersection numbers of $Γ$ are expressed in terms of $p$, $q$, and $r$, where $r$ is the size of the antipodal classes of $Γ$. We denote $Γ$ by $\mathrm{AT4}(p,q,r)$ and call this an antipodal tight graph of diameter $4$ with parameters $p,q,r$. In this paper, we give a new feasibility condition for the $\mathrm{AT4}(p,q,r)$ family. We determine a necessary and sufficient condition for the second subconstituent of $\mathrm{AT4}(p,q,2)$ to be an antipodal tight graph. Using this condition, we prove that there does not exist $\mathrm{AT4}(q^3-2q,q,2)$ for $q\equiv3$ $(\mathrm{mod}~4)$. We discuss the $\mathrm{AT4}(p,q,r)$ graphs with $r=(p+q^3)(p+q)^{-1}$. |
| title | A New Feasibility Condition for the AT4 Family |
| topic | Combinatorics 05E30 |
| url | https://arxiv.org/abs/2204.07842 |