Genus and crosscap of solvable conjugacy class graphs of finite groups

The solvable conjugacy class graph of a finite group G, denoted by \(\Gamma _{sc}(G)\) ${\mathrm{\Gamma }}_{\mathit{sc}}(G)$ , is a simple undirected graph whose vertices are the non-trivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist \(x \in C\) $x\in C$ and \(y \in D\) $y\in D$ such that \(\langle x, y\rangle \) $⟨x,y⟩$ is solvable. In this paper, we discuss certain properties of the genus and crosscap of \(\Gamma _{sc}(G)\) ${\mathrm{\Gamma }}_{\mathit{sc}}(G)$ for the groups \(D_{2n}\) $D_{2n}$ , \(Q_{4n}\) $Q_{4n}$ , \(S_n\) $S_n$ , \(A_n\) $A_n$ , and \({{\,\mathrm{\mathop {\textrm{PSL}}}\,}}(2,2^d)\) $\text{PSL}(2,2^d)$ . In particular, we determine all positive integers n such that their solvable conjugacy class graphs are planar, toroidal, double-toroidal, or triple-toroidal. We shall also obtain a lower bound for the genus of \(\Gamma _{sc}(G)\) ${\mathrm{\Gamma }}_{\mathit{sc}}(G)$ in terms of the order of the center and number of conjugacy classes for certain groups. As a consequence, we shall derive a relation between the genus of \(\Gamma _{sc}(G)\) ${\mathrm{\Gamma }}_{\mathit{sc}}(G)$ and the commuting probability of certain finite non-solvable group.