The power graph on the conjugacy classes of a finite group (Q519947)

Let \(G\) be a group. The vertices of the digraph \(\vec{{\mathcal P}}_{\mathcal C}(G)\) are the non-trivial conjugacy classes of \(G\) and there is an edge from \(C\) to \(C'\) if \(C\neq C'\) and \(C \subseteq (C')^m\) for some positive \(m\) (or equivalently, if \(C \subseteq \left\langle C' \right\rangle\)). The vertices of the graph \({\mathcal P}_{\mathcal C}(G)\) are all conjugacy classes of \(G\) and there is an edge between \(C \neq C'\) if one is a subset of the power of the other. The author characterizes when the digraph \(\vec{{\mathcal P}}_{\mathcal C}(G)\) or the graph \({\mathcal P}_{\mathcal C}(G)\) is complete, or regular, or isomorphic to a star graph or to a wheel graph.