Genus and crosscap of solvable conjugacy class graphs of finite groups

Abstract

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.