Genus of commuting conjugacy class graph of certain finite groups

For a non-abelian group G, its commuting conjugacy class graph \mathcal{CCC}(G) is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of G and two distinct vertices x^G and y^G are adjacent if there exists some elements x' \in x^G and y' \in y^G such that x'y' = y'x'. In this paper we compute the genus of \mathcal{CCC}(G) for six well-known classes of non-abelian two-generated groups (viz. D_{۲n}, SD_{۸n}, Q_{۴m}, V_{۸n}, U_{(n, m)} and G(p, m, n)) and determine whether \mathcal{CCC}(G) for these groups are planar, toroidal, double-toroidal or triple-toroidal.

Keywords:

Commuting conjugacy class graph , Genus , Finite group