Let G be a finite group and V(G) be the set of all non-central conjugacy classes of G. Consider Γ(G) of G which is defined as follows: its vertex set is the set V(G) and two distinct vertices x^G and y^G are connected with an edge if o(x) and o(y) are not coprime. In this article, we determine the structure of groups G whose non-trivial elements have prime orders and whose graph Γ(G) does not have any complete subgraph with six vertices
