In this paper we characterize groups according to the number of end vertices in the associated coprime graphs. An upper bound on the order of the group that depends on the number of end vertices is obtained. We also prove that \(2-\)groups are the only groups whose coprime graphs have odd number of end vertices. Classifications of groups with small number of end vertices in the coprime graphs are given. We give a complete answer to [4, Question 3.7], where we show that \(\mathbb {Z}_4\) and \(\mathbb {Z}_2\times \mathbb {Z}_2\) are the only groups whose coprime graph has exactly three end vertices.

中文翻译：

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根据互质图中的端点数对组进行分类

在本文中，我们根据相关互质图中的端点数来表征组。获得取决于末端顶点数的组阶数的上限。我们还证明了\(2-\)群是唯一的互质图具有奇数个端点的群。给出了互质图中末端顶点数少的群的分类。我们给出了 [4, Question 3.7] 的完整答案，其中我们证明\(\mathbb {Z}_4\)和\(\mathbb {Z}_2\times \mathbb {Z}_2\)是仅有的组其互质图正好有三个端点。