Groups whose prime graph on conjugacy class sizes has few complete vertices

Let G be a finite group, and let Γ(G) denote the prime graph built on the set of conjugacy class sizes of G. In this paper, we consider the situation when Γ(G) has “few complete vertices”, and our aim is to investigate the influence of this property on the group structure of G. More precisely, assuming that there exists at most one vertex of Γ(G) that is adjacent to all the other vertices, we show that G is solvable with Fitting height at most 3 (the bound being the best possible). Moreover, if Γ(G) has no complete vertices, then G is a semidirect product of two abelian groups having coprime orders. Finally, we completely characterize the case when Γ(G) is a regular graph.