The coprime graph of a group

The coprime graph $gg$ with a finite group $G$ as follows: Take $G$ as the vertices of $gg$ and join two distinct vertices $u$ and $v$ if $(|u|,|v|)=1$. In this paper, we want to explore how the graph theoretical properties of $gg$ can effect on the group theoretical properties of $G$. First, we investigate some properties of $gg$ and characterize certain finite groups whose the coprime graph have some property. Then, we discuss some interesting questions with the automorphism groups of coprime graphs. Finally, we prove some general graph theoretical properties of the coprime graphs of the dihedral groups.