Abstract
In this paper we study the coprime graph of a group $G$. The coprime graph of a group $G$, denoted by $Gamma_G$, is a graph whose vertices are elements of $G$ and two distinct vertices $x$ and $y$ are adjacent if and only if $(|x|,|y|)=1$. In this paper, we show that $chi(Gamma_G)=omega(Gamma_G).$ We classify all the groups which $Gamma_G$ is a complete $r-$partite graph or a planar graph. Also we study the automorphism group of $Gamma_G$.
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