On combinatorial properties of Gruenberg--Kegel graphs of finite groups
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arXiv
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| Format: | Preprint |
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2024
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| author | Chen, Mingzhu Gorshkov, Ilya B. Maslova, Natalia V. Yang, Nanying |
| author_facet | Chen, Mingzhu Gorshkov, Ilya B. Maslova, Natalia V. Yang, Nanying |
| contents | If $G$ is a finite group, then the spectrum $ω(G)$ is the set of all element orders of $G$. The prime spectrum $π(G)$ is the set of all primes belonging to $ω(G)$. A simple graph $Γ(G)$ whose vertex set is $π(G)$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if $rs \in ω(G)$ is called the Gruenberg-Kegel graph or the prime graph of $G$.
In this paper, we prove that if $G$ is a group of even order, then the set of vertices which are non-adjacent to $2$ in $Γ(G)$ form a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg-Kegel graph of a finite group. Besides this, we prove that a complete bipartite graph with each part of size at least $3$ can not be isomorphic to the Gruenberg-Kegel graph of a finite group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_04789 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On combinatorial properties of Gruenberg--Kegel graphs of finite groups Chen, Mingzhu Gorshkov, Ilya B. Maslova, Natalia V. Yang, Nanying Group Theory Combinatorics 20D60, 05C25 If $G$ is a finite group, then the spectrum $ω(G)$ is the set of all element orders of $G$. The prime spectrum $π(G)$ is the set of all primes belonging to $ω(G)$. A simple graph $Γ(G)$ whose vertex set is $π(G)$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if $rs \in ω(G)$ is called the Gruenberg-Kegel graph or the prime graph of $G$. In this paper, we prove that if $G$ is a group of even order, then the set of vertices which are non-adjacent to $2$ in $Γ(G)$ form a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg-Kegel graph of a finite group. Besides this, we prove that a complete bipartite graph with each part of size at least $3$ can not be isomorphic to the Gruenberg-Kegel graph of a finite group. |
| title | On combinatorial properties of Gruenberg--Kegel graphs of finite groups |
| topic | Group Theory Combinatorics 20D60, 05C25 |
| url | https://arxiv.org/abs/2401.04789 |