On transitive permutation groups with exponential graph growth
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916905030254592 |
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| author | Mitrović, Đorđe Verret, Gabriel |
| author_facet | Mitrović, Đorđe Verret, Gabriel |
| contents | Let $Γ$ be a finite connected graph and $G$ a vertex-transitive group of its automorphisms. The pair $(Γ, G)$ is said to be locally-$L$ if the permutation group induced by the action of the vertex-stabiliser $G_v$ on the set of neighbours of a vertex $v$ in $Γ$ is permutation isomorphic to $L$. The maximum growth of $|G_v|$ as a function of $|VΓ|$ for locally-$L$ pairs $(Γ,G)$ is called the graph growth of $L$. We prove that if $L$ is a transitive permutation group on a set $Ω$ admitting a nontrivial block $B$ such that the pointwise stabiliser of $Ω\setminus B$ in $L$ is nontrivial, then the graph growth of $L$ is exponential. This generalises several results in the literature on transitive permutation groups with exponential graph growth. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12588 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On transitive permutation groups with exponential graph growth Mitrović, Đorđe Verret, Gabriel Combinatorics Group Theory 20B25, 05E18 Let $Γ$ be a finite connected graph and $G$ a vertex-transitive group of its automorphisms. The pair $(Γ, G)$ is said to be locally-$L$ if the permutation group induced by the action of the vertex-stabiliser $G_v$ on the set of neighbours of a vertex $v$ in $Γ$ is permutation isomorphic to $L$. The maximum growth of $|G_v|$ as a function of $|VΓ|$ for locally-$L$ pairs $(Γ,G)$ is called the graph growth of $L$. We prove that if $L$ is a transitive permutation group on a set $Ω$ admitting a nontrivial block $B$ such that the pointwise stabiliser of $Ω\setminus B$ in $L$ is nontrivial, then the graph growth of $L$ is exponential. This generalises several results in the literature on transitive permutation groups with exponential graph growth. |
| title | On transitive permutation groups with exponential graph growth |
| topic | Combinatorics Group Theory 20B25, 05E18 |
| url | https://arxiv.org/abs/2508.12588 |