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In this paper, we study a graph whose vertices are the non-trivial conjugacy classes of a finite group, and two conjugacy classes  C1  and  C2  are adjacent if and only if we can findandsuch that  x ,  y  commute. In particular, we investigate groups whose corresponding graphs contain isolated vertices. We classify solvable groups with this property and show that if  G  is non-solvable, then eitheris almost simple or  G  is sharply 2-transitive. We determine almost simple groups with sporadic socles and some families of simple groups with this property.

Finite groups whose commuting conjugacy class graphs have isolated vertices