### On the dominating number, independent number and the regularity of the relative co-prime graph of a group

#### Abstract

Let *H* be a subgroup of a finite group *G*. The co-prime graph of a group is defined as a graph whose vertices are elements of *G* and two distinct vertices* *are adjacent if and only if the greatest common divisor of order of *x* and *y* is equal to one. This concept has been extended to the relative co-prime graph of a group with respect to a subgroup *H*, which is defined as a graph whose vertices are elements of *G* and two distinct vertices *x* and *y* are joined by an edge if and only if their orders are co-prime and any of *x* or *y* is in *H*. Some properties of graph such as the dominating number, degree of a dominating set of order one and independent number are obtained. Lastly, the regularity of the relative co-prime graph of a group is found.

#### Keywords

#### Full Text:

PDF#### References

Abd Rhani, N. Mohd Ali, N. M., Sarmin, N. H., Erfanian, A. 2017. The relative co-prime graph of a group. Proc. of the Fourth Biennial Int. Group Theory Conference 2017. 155-158.

Bondy, J. A., Murty, U. S. R. 1982. Graphs theory with applications. North Holand, New York, Amsterdam, Oxford: Elsevier Science. Retrieved from https://www.iro.umontreal.ca/~hahn/IFT3545/GTWA.pdf.

Doostabadi, A., Erfanian, A., Jafarzadeh, A. 2015. Some results on the power graphs of finite groups. ScienceAsia. 41, 73–78.

Godsil, C., Royle, G. F. 2001. Algebraic graph theory (5th edition). Boston, New York:Springer-Verlag.Retrieved from http://www.springer.com/cn/book/9780387952413.

Harary, F. 1965. Graph theory. California, London, Ontario: Addison-Wesley. Retrieved from http://www.dtic.mil/dtic/tr/fulltext/u2/705364.pdf.

Iiyori, N., Yamaki, H. 1993. Prime graph components of the simple groups of Lie type over the field of even characteristic. J. Algebra. 155, 335–343.

Ma, X. L., Wei, H. Q., Yang, L. Y. 2014. The coprime graph of a group, Int. J. Group Theory. 3, 13–23.

Rajkumar, R and Devi, P. 2015. Coprime graph of subgroups of a group, arXiv:1510.001129v2 [math.GR].

Tamizh Chelvam, T., Sattanathan, M. 2013. Power graph of finite abelian groups. J. Algebra and Discrete Mathematics. 16, 33-41.

Williams, J. S. 1981. Prime graph components of finite groups. J. Algebra. 69, 487-513.

DOI: http://dx.doi.org/10.11113/mjfas.v13n2.602

### Refbacks

- There are currently no refbacks.

Copyright (c) 2017 Norarida Abd Rhani, Nor Muhainiah Mohd Ali, Nor Haniza` Sarmin, Ahmad Erfanian

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Copyright © 2016 Penerbit UTM Press, Universiti Teknologi Malaysia. Disclaimer: This website has been updated to the best of our knowledge to be accurate. However, Universiti Teknologi Malaysia shall not be liable for any loss or damage caused by the usage of any information obtained from this website.