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

• Main Authors: Abd. Rhani, N., Mohd. Ali, N. M., Sarmin, N. H., Erfanian, A.
• Format: Article
• Published: Penerbit UTM Press 2017
• Subjects: QA Mathematics
• Online Access: http://eprints.utm.my/id/eprint/80920/; http://dx.doi.org/10.11113/mjfas.v13n2.602

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.