On a Graph Associated to Groups

Let G be a group such that the subgroup T^2(G)={x∈G|(gx)^2=(xg)^2, for all g∈G} is a proper subgroup. We define a graph Γ with the vertex set G∖T^2(G) in which two vertices x,y are joined by an edge if (xy)^2≠(yx)^2. In this note we study this graph and find a characterization of some dihedral groups in terms of this graph.