  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. One of the results shows that $\mathbb{Z}_4$ and
$\mathbb{Z}_2\times \mathbb{Z}_2$ are the only groups whose coprime graph has
exactly three end vertices.
