A classification of $Q$-polynomial distance-regular graphs with girth $6$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929685335638016 |
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| author | Miklavič, Štefko |
| author_facet | Miklavič, Štefko |
| contents | Let $Γ$ denote a $Q$-polynomial distance-regular graph with diameter $D$ and valency $k \ge 3$. In [Homotopy in $Q$-polynomial distance-regular graphs, Discrete Math., {\bf 223} (2000), 189-206], H. Lewis showed that the girth of $Γ$ is at most $6$. In this paper we classify graphs that attain this upper bound. We show that $Γ$ has girth $6$ if and only if it is either isomorphic to the Odd graph on a set of cardinality $2D +1$, or to a generalized hexagon of order $(1, k -1)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12820 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A classification of $Q$-polynomial distance-regular graphs with girth $6$ Miklavič, Štefko Combinatorics 05C50, 05C38 Let $Γ$ denote a $Q$-polynomial distance-regular graph with diameter $D$ and valency $k \ge 3$. In [Homotopy in $Q$-polynomial distance-regular graphs, Discrete Math., {\bf 223} (2000), 189-206], H. Lewis showed that the girth of $Γ$ is at most $6$. In this paper we classify graphs that attain this upper bound. We show that $Γ$ has girth $6$ if and only if it is either isomorphic to the Odd graph on a set of cardinality $2D +1$, or to a generalized hexagon of order $(1, k -1)$. |
| title | A classification of $Q$-polynomial distance-regular graphs with girth $6$ |
| topic | Combinatorics 05C50, 05C38 |
| url | https://arxiv.org/abs/2501.12820 |