Genus and crosscap of solvable conjugacy class graphs of finite groups

The solvable conjugacy class graph of a finite group G, denoted by \(\Gamma _{sc}(G)\), 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 \(x \in C\) and \(y \in D\) such that \(\langle x, y\rangle \) is solvable. In this paper, we discuss certain properties of the genus and crosscap of \(\Gamma _{sc}(G)\) for the groups \(D_{2n}\), \(Q_{4n}\), \(S_n\), \(A_n\), and \({{\,\mathrm{\mathop {\textrm{PSL}}}\,}}(2,2^d)\). 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 \(\Gamma _{sc}(G)\) 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 \(\Gamma _{sc}(G)\) and the commuting probability of certain finite non-solvable group.