# Parameters of the coprime graph of a group

Document Type : Research Paper

Authors

Department of Mathematics, Winthrop University, 142 Bancroft Hall Rock Hill, SC, USA

Abstract

‎There are many different graphs one can associate to a group‎. ‎Some examples are the well-known Cayley graph‎, ‎the zero divisor graph (of a ring)‎, ‎the power graph‎, ‎and the recently introduced coprime graph of a group‎. ‎The coprime graph of a group $G$‎, ‎denoted $\Gamma_G$‎, ‎is the graph whose vertices are the group elements with $g$ adjacent to $h$ if and only if $(o(g),o(h))=1$‎. ‎In this paper we calculate the independence number of the coprime graph of the dihedral groups‎. ‎Additionally‎, ‎we characterize the groups whose coprime graph is perfect‎.

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#### References

[1] D. Anderson and Ph. Livingston, The zero-divisor graph of a commutative ring, J. Algebra, 217 (1999) 434–447.
[2] G. Chartrand and P. Zhang, A First Course in Graph Theory, Dover Publications, Inc., Mineola, NY, 2012.
[3] M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The strong perfect graph theorem, Ann. of Math. (2),
164 (2006) 51–229.
[4] S. Curran and J. Gallian, Hamiltonian cycles and paths in Cayley graphs and digraphs – a survey. Discrete Math.,
156 (1996) 1–18.
[5] H. Dorbidi, A note on the coprime graph of a group, Int. J. Group Theory, 5 (2016) 17–22.
[6] H. R. Dorbidi, Independent sets in the coprime graph of a group, 10th Graph Theory and Algebraic Combinatorics
Conference of Iran, Yazd University, January, 2018 17–18.
[7] J. Gallian, Contemporary Abstract Algbera, Sixth Edition, Houghton Mifflin Company, Boston, MA, 2006.
[8] X. Long Ma, H. Quan Wei and Li Ying Yang, The coprime graph of a group, Int. J. Group Theory, 3 (2014) 13–23.
[9] Sh. Rehman, A. Baig, M. Imran and Z. Khan, Order divisor graphs of finite groups, arXiv:1611.04280 [math.CO].
[10] K. Selvakumar and M. Subajini, Classification of Groups with Toroidal Coprime Graphs, Australas. J. Combin., 69
(2017) 174–183.