### Journals Information

**
Mathematics and Statistics Vol. 10(2), pp. 436 - 441 DOI: 10.13189/ms.2022.100219 Reprint (PDF) (275Kb) **

## The Non-Abelian Tensor Square Graph Associated to a Symmetric Group and its Perfect Code

**Athirah Zulkarnain ^{1}^{,*}, Hazzirah Izzati Mat Hassim ^{1}, Nor Haniza Sarmin ^{1}, Ahmad Erfanian ^{2}**

^{1}Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia

^{2}Department of Pure Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

**ABSTRACT**

A set of vertices and edges forms a graph. A graph can be associated with groups using the groups' properties for its vertices and edges. The set of vertices of the graph comprises the elements of the group, while the set of edges of the graph is the properties and requirements for the graph. A non-abelian tensor square graph of a group is defined when its vertex set represents the non-tensor centre elements' set of G. Then, two distinguished vertices are connected by an edge if and only if the non-abelian tensor square of these two elements is not equal to the identity of the non-abelian tensor square. This study investigates the non-abelian tensor square graph for the symmetric group of order six. In addition, some properties of this group's non-abelian tensor square graph are computed, including the diameter, the dominating number, and the chromatic number. The perfect code for the non-abelian tensor square graph for a symmetric group of order six is also found in this paper.

**KEYWORDS**

Graph Theory, Non-Abelian Tensor Square, Chromatic Number, Diameter

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Athirah Zulkarnain , Hazzirah Izzati Mat Hassim , Nor Haniza Sarmin , Ahmad Erfanian , "The Non-Abelian Tensor Square Graph Associated to a Symmetric Group and its Perfect Code," Mathematics and Statistics, Vol. 10, No. 2, pp. 436 - 441, 2022. DOI: 10.13189/ms.2022.100219.

(b). APA Format:

Athirah Zulkarnain , Hazzirah Izzati Mat Hassim , Nor Haniza Sarmin , Ahmad Erfanian (2022). The Non-Abelian Tensor Square Graph Associated to a Symmetric Group and its Perfect Code. Mathematics and Statistics, 10(2), 436 - 441. DOI: 10.13189/ms.2022.100219.