Regular character degree graphs

Let Γ be a character degree graph with n vertices of some finite solvable group G. If Γ is regular, we prove that Γ   is either complete or (n−2)(n−2)-regular. Moreover, if Γ   is (n−2)(n−2)-regular and G has no normal non-abelian Sylow subgroups, we show that G is a direct product of groups having disconnected character degree graph.