# Genus of commuting conjugacy class graph of certain finite groups

Document Type : Research Paper

Authors

1 Department of Mathematical Sciences, Tezpur University, Sonitpur, India

2 Department of Mathematical Sciences, Tezpur University, Napaam-784028, Sonitpur, Assam, India. Department of Mathematics, Cachar College, Silchar-788001, Assam, India.

10.29252/as.2021.2444

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.

Keywords

#### References

[1] M. Afkhami, M. Farrokhi D. G. and K. Khashyarmanesh, Planar, toroidal, and projective commuting and non-commuting graphs, Comm. Algebra, 43 No. 7 (2015) 2964-2970.
[2] P. Bhowal and R. K. Nath, Spectral aspects of commuting conjugacy class graph of finite groups, Alg. Struc. Appl., 8 No. 2 (2021) 67-118.
[3] P. Bhowal and R. K. Nath, Spectrum and energies of commuting conjugacy class graph of a finite group, available at https://arxiv.org/abs/2003.07142.
[4] P. Bhowal, D. Nongsiang and R. K. Nath, Solvable graphs of finite groups, Hacet. J. Math. Stat., 49 No. 6 (2020) 1955-1964.
[5] R. Brauer and K. A. Fowler, On groups of even order, Ann. Math., 62 No. 2 (1955) 565-583.
[6] A. K. Das and D. Nongsiang, On the genus of the commuting graphs of finite non-abelian groups, Int. Electron. J. Algebra, 19 (2016) 91-109.
[7] A. K. Das and D. Nongsiang, On the genus of the nilpotent graphs of finite groups Comm. Algebra, 43 No. 1 (2015) 5282-5290.
[8] M. Herzog, M. Longobardi and M. Maj, On a commuting graph on conjugacy classes of groups, Comm. Algebra, 37 No. 10 (2009) 3369-3387.
[9] A. Mohammadian, A. Erfanian, D. G. M. Farrokhi and B. Wilkens, Triangle-free commuting conjugacy class graphs, J. Group Theory, 19 (2016) 1049-1061.
[10] M. A. Salahshour and A. R. Ashrafi, Commuting conjugacy class graph of finite CA-groups, Khayyam J. Math., 6 No. 1 (2020) 108-118.
[11] J. Battle, F. Harary, Y. Kodama and J. W. T. Youngs, Additivity of the genus of a graph, Bull. Amer. Math. Soc., 68 No. 6 (1962) 565-568.
[12] A. T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, No. 8., American Elsevier Publishing Co., Inc., New York, 1973.