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

Norarida Abd Rhani, Nor Muhainiah Mohd Ali, Nor Haniza Sarmin, Ahmad Erfanian

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


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

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References


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

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