The conjugacy classes of a group play an important role in its structure. The authors consider the set of non-central conjugacy classes of a finite group \(G\), \(\hbox{Con}(G)\). They then construct a graph \(\Gamma(G)\) whose vertices are in 1-1 correspondence with \(\hbox{Con}(G)\) and where two vertices are connected when the sizes of the conjugacy classes are not coprime. They prove two interesting theorems about this graph. I will state this as just one theorem. Theorem: Let \(G\) be a finite group. Then (a) \(\Gamma(G)\) has at most two components and (b) \(\Gamma(G)\) has two components if and only if \(G/Z(G)\) is a Frobenius group such that the preimages of both the kernel and the complement are Abelian. In the final section they extend these results to FC-groups.