论有限群的格伦伯格-凯格尔图的组合特性

Mingzhu Chen, Ilya Gorshkov, Natalia V. Maslova, Nanying Yang
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引用次数: 0

摘要

如果 G 是一个有限群,那么谱 \(\omega (G)\) 是 G 的所有元素阶的集合。素数谱 \(\pi (G)\) 是属于 \(\omega (G)\) 的所有素数的集合。一个简单的图(\(\Gamma (G)\))的顶点集合是(\(\pi (G)\)),并且其中两个不同的顶点 r 和 s 相邻,当且仅当(\(rs \in \omega (G)\))是且仅当(\(rs \in \omega (G)\)),这个图被称为格伦伯格-凯格尔图或 G 的素数图。在本文中,我们证明了如果 G 是偶数阶群,那么在 \(\Gamma (G)\) 中与 2 不相邻的顶点集合构成了小群的联合。此外,我们还确定了强规则图何时与有限群的格伦伯格-凯格尔图同构。
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On combinatorial properties of Gruenberg–Kegel graphs of finite groups

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

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