Document Type : Research Paper

**Authors**

Department of Mathematics, National Institute of Technology Calicut, P.O.Box 673601, Calicut, India

10.22108/toc.2024.137172.2055

**Abstract**

For a finite group $G$ the co-prime graph $\Gamma(G)$ is defined as a graph with vertex set $G$ in which two distinct vertices $x$ and $y$ are adjacent if and only if $gcd(o(x),o(y))=1$ where $o(x)$ and $o(y)$ denote the orders of the elements $x$ and $y$ respectively. In this paper we find properties of groups whose co-prime graphs forbid graphs such as $C_4,K_{1,3},P_4$ and asteroidal triples.

**Keywords**

**Main Subjects**

[2] A. G. Syarifudin, I. G. A. W. Wardhana, N. W. Switrayni and Q. Aini, The cique numbers and chromatic numbers of the coprime graph of a dihedral group, IOP Conference Series: Materials Science and Engineering, 1115 no. 1 (2021) 012083.

[3] A. Kelarev, Graph algebras and automata, Monographs and Textbooks in Pure and Applied Mathematics, 257, Marcel Dekker, Inc., New York, 2003.

[4] H. B. Shelash and M. Jasim, Co-prime Graph of Finite Groups, Order, 1 (2021) 2n.

[5] H. R. Dorbidi, A note on the coprime graph of a group, Int. J. Group Theory, 5 no. 4 (2016) 17–22.

[6] M. Saini, S. Khasraw and M. S. Sanhan, On co-prime order graphs of finite abelian p-groups, J. Math. Comput. Sci., 11 no. 6 (2021) 7052–7061.

[7] N. Nurhabibah, A. G. Syarifudin and I. G. A. W. Wardhana, Some results of the coprime graph of a generalized quaternion group Q 4n, InPrime: Indonesian Journal of Pure and Applied Mathematics, 3 no. 1 (2021) 29–33.

[8] P. Erdös, A. W. Goodman and L. Pósa, The representation of a graph by set intersections, Canadian J. Math., 18 (1966) 106–112.

[9] P. J. Cameron, P. Manna and R. Mehatari, On finite groups whose power graph is a cograph, J. Algebra, 591 (2022) 59–74.

[10] P. Manna, P. J. Cameron and R. Mehatari, Forbidden subgraphs of power graphs, Electron. J. Combin., 28 no. 3 (2021) 14 pp.

[11] R. Juliana, M. Masriani, I. G. A. W. Wardhana, N. W. Switrayni and I. Irwansyah, Coprime graph of integer modulo n group and its subgroups, Journal of Fundamental Mathematics and Applications (JFMA), 3 no. 1 (2020) 15–18.

[12] S. Hao, G. Zhong and X. Ma, Notes on the co-prime order graph of a group, C. R. Acad. Bulgare Sci., 75 (2022) 340–348.

[13] X. Ma, H. Wei and L. Yang, The coprime graph of a group, Int. J. Group Theory, 3 no. 3 (2014) 13–23.

Articles in Press, Corrected Proof

Available Online from 18 August 2024

**Receive Date:**23 March 2023**Revise Date:**11 June 2024**Accept Date:**15 June 2024**Published Online:**18 August 2024