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In this paper we characterize groups according to the number of end verticesin the associated coprime graphs. An upper bound on the order of the group thatdepends 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 endvertices. Classifications of groups with small number of end vertices in thecoprime graphs are given. One of the results shows that $\mathbb{Z}_4$ and$\mathbb{Z}_2\times \mathbb{Z}_2$ are the only groups whose coprime graph hasexactly three end vertices.

Classification of Groups according to the number of end vertices in the coprime graph