Abstract
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 find and
such 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 either
is 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.
2020 Mathematics Subject Classification:
Acknowledgments
The author is very much indebted to Professor Chris Parker for his helpful comments and suggestions which improved the results of this paper, especially by suggesting Proposition 3.9.