Let G be a finite group, and let cs(G) be the set of the sizes of the conjugacy A vertex of a graph is said to be complete if it is adjacent to all other vertices of the graph. In this thesis, we consider the situation when ? (G) has “few complete vertices”, and our aim is to investigate the influence of this property on the group structure of G. More precisely we show G as a finite group are that at most one vertex of ? (G) is complete. Then G is solvable, and the Fitting height of G is at most 3. The conclusion of the above fact is as follows: Let G be a finite solvable group such that ? (G) is of bounded Fitting height for conjugacy
