On the (non-)existence of tight distance-regular graphs: a local approach
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911871998623744 |
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| author | Koolen, Jack H. Lee, Jae-Ho Li, Shuang-Dong Li, Yun-Han Liang, Xiaoye Tan, Ying-Ying |
| author_facet | Koolen, Jack H. Lee, Jae-Ho Li, Shuang-Dong Li, Yun-Han Liang, Xiaoye Tan, Ying-Ying |
| contents | Let $Γ$ denote a distance-regular graph with diameter $D\geq 3$. Jurišić and Vidali conjectured that if $Γ$ is tight with classical parameters $(D,b,α,β)$, $b\geq 2$, then $Γ$ is not locally the block graph of an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices $x, y, z$ of $Γ$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both $x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We then show that if $Γ$ is locally the block graph of an orthogonal array (resp. a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this result, we prove that if a tight distance-regular graph $Γ$ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of $Γ$ is bounded by a function in the parameter $b=b_1/(1+θ_1)$, where $b_1$ is the intersection number of $Γ$ and $θ_1$ is the second largest eigenvalue of $Γ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_05595 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the (non-)existence of tight distance-regular graphs: a local approach Koolen, Jack H. Lee, Jae-Ho Li, Shuang-Dong Li, Yun-Han Liang, Xiaoye Tan, Ying-Ying Combinatorics 05E30 Let $Γ$ denote a distance-regular graph with diameter $D\geq 3$. Jurišić and Vidali conjectured that if $Γ$ is tight with classical parameters $(D,b,α,β)$, $b\geq 2$, then $Γ$ is not locally the block graph of an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices $x, y, z$ of $Γ$, where $x$ and $y$ are adjacent, and $z$ is at distance $2$ from both $x$ and $y$, the number of common neighbors of $x$, $y$, $z$ is constant. We then show that if $Γ$ is locally the block graph of an orthogonal array (resp. a Steiner system) with smallest eigenvalue $-m$, $m\geq 3$, then the intersection number $c_2$ is not equal to $m^2$ (resp. $m(m+1)$). Using this result, we prove that if a tight distance-regular graph $Γ$ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of $Γ$ is bounded by a function in the parameter $b=b_1/(1+θ_1)$, where $b_1$ is the intersection number of $Γ$ and $θ_1$ is the second largest eigenvalue of $Γ$. |
| title | On the (non-)existence of tight distance-regular graphs: a local approach |
| topic | Combinatorics 05E30 |
| url | https://arxiv.org/abs/2312.05595 |