On combinatorial properties of Gruenberg–Kegel graphs of finite groups

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发表日期：

2024年8月20日

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Mingzhu Chen,
Ilya Gorshkov,
Natalia V. Maslova,
Nanying Yang

摘要

If G is a finite group, then the spectrum $\omega (G)$ $ω (G)$ is the set of all element orders of G. The prime spectrum $\pi (G)$ $π (G)$ is the set of all primes belonging to $\omega (G)$ $ω (G)$ . A simple graph $\Gamma (G)$ $Γ (G)$ whose vertex set is $\pi (G)$ $π (G)$ and in which two distinct vertices r and s are adjacent if and only if $rs \in \omega (G)$ $r s \in ω (G)$ 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 $\Gamma (G)$ $Γ (G)$ forms a union of cliques. Moreover, we decide when a strongly regular graph is isomorphic to the Gruenberg–Kegel graph of a finite group.