Solvable conjugacy class graph of groups (Q6041569)

The authors introduce a graph \(\Gamma_{\mathrm{sc}}(\mathrm{G})\) associated with a group \(\mathrm{G}\), called the solvable conjugacy class graph (abbreviated as \(\mathrm{SCC}\)-graph), whose vertices are the nontrivial conjugacy classes of \(\mathrm{G}\). Two distinct conjugacy classes \(\mathrm{C, D}\) are adjacent if there exist \(x \in\mathrm{C}\) and \(y \in\mathrm{D}\) such that \(\langle x,y \rangle\) is solvable. The authors discuss the connectivity, girth, clique number, and several other properties of the \(\mathrm{SCC}\)-graph. They show that there are only finitely many finite groups whose \(\mathrm{SCC}\)-graph has given clique number \(\mathrm{d}\). They list such groups with \(\mathrm{d} = 2\).