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

## DOI:

https://doi.org/10.11113/mjfas.v13n2.602## 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.

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