^{}Department of Basic Sciences, University of Jiroft, Jiroft, Kerman, Iran

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 $\Gamma_G$.