On some graphs related to regular, oriented triangular maps (Q1362995)

Let \(G\) be a finite group and \(\lambda\) a conjugacy class of \(G\). Then \(\psi(G,\lambda)\) is defined to be a graph with vertex set \(\lambda\) and two distinct vertices \(u\), \(v\) being adjacent iff both \(uv^2\) and \(vu^2\) are involutions. Under certain conditions on the pair \((G,\lambda)\), the graph \(\psi(G,\lambda)\) defines a second graph \(\xi(G,\lambda)\) which consists of the disjoint union of disjoint copies of the underlying graph \(\theta_i\) of \({\mathcal M}_i\), where \({\mathcal M}_i\) is a regular oriented triangular map.