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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.01075 |
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| _version_ | 1866910350584053760 |
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| author | Huang, Jun-Jie Feng, Yan-Quan Zhou, Jin-Xin Yin, Fu-Gang |
| author_facet | Huang, Jun-Jie Feng, Yan-Quan Zhou, Jin-Xin Yin, Fu-Gang |
| contents | A vertex transitive graph $Γ$ is said to be $2$-distance transitive if for each vertex $u$, the group of automorphisms of $Γ$ fixing the vertex $u$ acts transitively on the set of vertices at distance $1$ and $2$ from $u$, while $Γ$ is said to be $2$-arc transitive if its automorphism group is transitive on the set of $2$-arcs. Then $2$-arc transitive graphs are $2$-distance transitive. The classification of $2$-arc transitive Cayley graphs on dihedral groups was given by Du, Malnič and Marušič in [Classification of 2-arc-transitive dihedrants, J. Combin. Theory Ser. B 98 (2008), 1349--1372]. In this paper, it is shown that a connected 2-distance transitive Cayley graph on the dihedral group of order $2n$ is either $2$-arc transitive, or isomorphic to the complete multipartite graph $K_{m[b]}$ for some $m\geq3$ and $b\geq2$ with $mb=2n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_01075 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The classification of two-distance transitive dihedrants Huang, Jun-Jie Feng, Yan-Quan Zhou, Jin-Xin Yin, Fu-Gang Combinatorics 05C25, 20B15, 20B30 A vertex transitive graph $Γ$ is said to be $2$-distance transitive if for each vertex $u$, the group of automorphisms of $Γ$ fixing the vertex $u$ acts transitively on the set of vertices at distance $1$ and $2$ from $u$, while $Γ$ is said to be $2$-arc transitive if its automorphism group is transitive on the set of $2$-arcs. Then $2$-arc transitive graphs are $2$-distance transitive. The classification of $2$-arc transitive Cayley graphs on dihedral groups was given by Du, Malnič and Marušič in [Classification of 2-arc-transitive dihedrants, J. Combin. Theory Ser. B 98 (2008), 1349--1372]. In this paper, it is shown that a connected 2-distance transitive Cayley graph on the dihedral group of order $2n$ is either $2$-arc transitive, or isomorphic to the complete multipartite graph $K_{m[b]}$ for some $m\geq3$ and $b\geq2$ with $mb=2n$. |
| title | The classification of two-distance transitive dihedrants |
| topic | Combinatorics 05C25, 20B15, 20B30 |
| url | https://arxiv.org/abs/2403.01075 |