# On the generalized commuting and non-commuting graph for metacyclic 3-groups

## Authors

• Siti Norziahidayu Amzee Zamri Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor
• Nor Haniza Sarmin Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor
• Mustafa Anis El-Sanfaz Department of Mathematics, Faculty of Science, University of Benghazi, Libya
• Hamisan Rahmat Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor

## Keywords:

Metacyclic 3-Groups, Commuting Graph, Non-Commuting Graph, Commute

## Abstract

Let   be a metacyclic 3-group and let   be a non-empty subset of   such that  . The generalized commuting and non-commuting graphs of a group   is denoted by   and   respectively. The vertex set of the generalized commuting and non-commuting graphs are the non-central elements in the set   such that     where   Two vertices in   are joined by an edge if they commute, meanwhile, the vertices in   are joined by an edge if they do not commute.

## References

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Omer, S. M. S., Sarmin, N. H., Erfanian, A., Moradipour, K. 2013. The probability that an element of a group fixes a set and and the group act on a set by conjugation. International Journal of Applied Mathematics and Statistics. 32(2), 111-117.

Goodman, F. M. 1998. Algebra: Abstract and Concrete. Upper Saddle River, N.J.: Prentice Hall.

El-Sanfaz, M. A. 2016. The Probability that an element of a non-abelian group fixes a set and its applications in graph theory. Doctor Philosophy, Universiti Teknologi Malaysia.