Genus of commuting conjugacy class graph of groups

TL;DR

This paper calculates the genus of the commuting conjugacy class graph for six well-known non-abelian two-generated groups, classifying their graphs as planar, toroidal, or higher genus surfaces.

Contribution

It provides explicit genus calculations for the commuting conjugacy class graphs of six specific classes of non-abelian groups, extending understanding of their topological properties.

Findings

Computed genus for each group class.

Classified graphs as planar, toroidal, or higher genus.

Identified topological types of these graphs.

Abstract

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. $D_{2n}, SD_{8n}, Q_{4m}, V_{8n}, U_{(n, m)}$ and $G(p, m, n)$) and determine whether $\mathcal{CCC}(G)$ for these groups are planar, toroidal, double-toroidal or triple-toroidal.