On two-distance-transitive graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916879108407296 |
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| author | Jin, Wei Koolen, Jack H. Lv, Chenhui |
| author_facet | Jin, Wei Koolen, Jack H. Lv, Chenhui |
| contents | A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and C. E. Praeger in 2012 about vertex-quasiprimitive $2$-distance-transitive graphs for the odd order case. Then we characterize $2$-distance-transitive graphs of valency $p$ or $p+1$ where $p$ is a prime. After that, as an application of the above result, we classify locally-primitive, $2$-distance-transitive graphs of small valency.
In addition to the above results on $2$-distance-transitive graphs, we also classify a family of amply regular graphs with diameter at least $4$ and parameters $(v, k, λ, \frac{k - 1}{2})$, and these graphs arise naturally in the classification of locally-primitive, $2$-distance-transitive graphs with small valency. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02010 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On two-distance-transitive graphs Jin, Wei Koolen, Jack H. Lv, Chenhui Combinatorics 05E18, 20B25 A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and C. E. Praeger in 2012 about vertex-quasiprimitive $2$-distance-transitive graphs for the odd order case. Then we characterize $2$-distance-transitive graphs of valency $p$ or $p+1$ where $p$ is a prime. After that, as an application of the above result, we classify locally-primitive, $2$-distance-transitive graphs of small valency. In addition to the above results on $2$-distance-transitive graphs, we also classify a family of amply regular graphs with diameter at least $4$ and parameters $(v, k, λ, \frac{k - 1}{2})$, and these graphs arise naturally in the classification of locally-primitive, $2$-distance-transitive graphs with small valency. |
| title | On two-distance-transitive graphs |
| topic | Combinatorics 05E18, 20B25 |
| url | https://arxiv.org/abs/2508.02010 |