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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/2403.01324 |
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| _version_ | 1866910350823129088 |
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| author | Fang, Teng Zhou, Sanming Zhou, Shenglin |
| author_facet | Fang, Teng Zhou, Sanming Zhou, Shenglin |
| contents | A graph $Γ$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $Γ$, where an arc is an ordered pair of adjacent vertices. Let $Γ$ be a $G$-symmetric graph such that its vertex set admits a nontrivial $G$-invariant partition ${\cal B}$, and let ${\cal D}(Γ, {\cal B})$ be the incidence structure with point set ${\cal B}$ and blocks $\{B\} \cup Γ_{\cal B}(α)$, for $B \in {\cal B}$ and $α\in B$, where $Γ_{\cal B}(α)$ is the set of blocks of ${\cal B}$ containing at least one neighbour of $α$ in $Γ$. In this paper we classify all $G$-symmetric graphs $Γ$ such that $Γ_{\cal B}(α) \ne Γ_{\cal B}(β)$ for distinct $α, β\in B$, the quotient graph of $Γ$ with respect to ${\cal B}$ is a complete graph, and ${\cal D}(Γ, {\cal B})$ is isomorphic to the complement of a $(G, 2)$-point-transitive linear space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01324 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A family of symmetric graphs in relation to 2-point-transitive linear spaces Fang, Teng Zhou, Sanming Zhou, Shenglin Combinatorics 05C25, 05E18 A graph $Γ$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $Γ$, where an arc is an ordered pair of adjacent vertices. Let $Γ$ be a $G$-symmetric graph such that its vertex set admits a nontrivial $G$-invariant partition ${\cal B}$, and let ${\cal D}(Γ, {\cal B})$ be the incidence structure with point set ${\cal B}$ and blocks $\{B\} \cup Γ_{\cal B}(α)$, for $B \in {\cal B}$ and $α\in B$, where $Γ_{\cal B}(α)$ is the set of blocks of ${\cal B}$ containing at least one neighbour of $α$ in $Γ$. In this paper we classify all $G$-symmetric graphs $Γ$ such that $Γ_{\cal B}(α) \ne Γ_{\cal B}(β)$ for distinct $α, β\in B$, the quotient graph of $Γ$ with respect to ${\cal B}$ is a complete graph, and ${\cal D}(Γ, {\cal B})$ is isomorphic to the complement of a $(G, 2)$-point-transitive linear space. |
| title | A family of symmetric graphs in relation to 2-point-transitive linear spaces |
| topic | Combinatorics 05C25, 05E18 |
| url | https://arxiv.org/abs/2403.01324 |