Character degree graphs that are complete graphs

Let G be a finite nonabelian group and let cd(G) denote the set whose elements are the (distinct) degrees of irreducible complex characters of G.The structure of G is deeply reflected and influenced by certain arithmetical properties of the set cd(G), and a useful way of studying such properties is to associate with cd(G) some particular graphs.One of these is the common divisor graph. The aim of this paper is to show that if the common divisor graph is a complete graph then G is soluble.