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

## Authors

• Norarida Abd Rhani Department of Mathematical Sciences, Faculty of Science,Universiti Teknologi Malaysia
• Nor Muhainiah Mohd Ali Department of Mathematical Sciences, Faculty of Science,Universiti Teknologi Malaysia
• Nor Haniza Sarmin Department of Mathematical Sciences, Faculty of Science,Universiti Teknologi Malaysia
• Ahmad Erfanian Department of Pure Mathematics, Faculty of Mathematical Sciences,Ferdowsi University of Mashhad

## Keywords:

Co-prime Graph, Relative Co-prime Graph, Dominating Number, Independent Number, Regular Graph

## 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.

## References

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