# Commuting Graphs, C(G, X) in Symmetric Groups Sym(n) and Its Connectivity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Commuting graph**denoted by $\mathcal{C}(G,X)$, is a graph whose set of vertices is X and two distinct vertices, $x,y\in X$, will be connected by an edge if the commutativity property is satisfied or $xy=yx$.

**Theorem**

**1.**

**Theorem**

**2.**

- 1.
- Γ is connected.
- 2.
- gcd$\left\{b\right(i):1\le i\le m\}=1$.
- 3.
- Γ has at least one edge which is not exact.
- 4.
- The vertex set of Γ is
**not**of the form $E\cup Y$, with $E\cap Y=\varnothing $ and $E,Y\ne \varnothing $, such that the following hold:- (a)
- for all $i,j\in E$ with $i\ne j,(i,j)$ is an exact edge;
- (b)
- there exists a vertex $y\in Y$ such that for all $i\in E,(i,y)$ is a special edge with source y;
- (c)
- no vertex of E is joined to a vertex of Y\$\left\{y\right\}$; and
- (d)
- gcd$\{b\left(i\right):i\in Y\}={e}_{y}$.

## 2. Connectedness of the Commuting Graph

**Theorem**

**3.**

- 1.
- $n=3r$ and $r=2$;
- 2.
- $n=3r+1$ or $n=3r+2$ and $r\ge 1$; and
- 3.
- $n=3r+q$; $r=1$ or $r=2$ and $q=3$.

**Proof**

**of**

**Theorem**

**3.**

**Condition**

**1.**

**Condition**

**2.**

**Condition**

**3.**

**Condition**

**4.**

- (a)
- Since there is only one vertex in E, we can join any two vertices neither by an exact edge nor by a non-exact edge.
- (b)
- Note that $(1,2)$ is the only edge of $\Gamma $ and it is not a special edge with source 2 since $b\left(2\right)=1\ne {e}_{2}$. Thus, Theorem 2 $\left(4\right)\left(b\right)$ is not satisfied.

## 3. Disconnected Commuting Graph and Its Connected Components

**Example**

**1.**

**Proposition**

**1.**

**Example**

**2.**

**Proposition**

**2.**

**Theorem**

**4.**

**Proof**

**of**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof**

**of**

**Theorem**

**5.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\mathcal{C}(G,X)$ | Commuting graph |

${C}_{G}\left(t\right)$ | Centralizer of an element t in group G |

${C}_{i}$ | i-th connected components |

Sym$\left(n\right)$ | Symmetric group of degree n |

lcm | Lowest common multiple |

gcd | Greatest common divisor |

## References

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**Figure 1.**Connected component ${C}_{1}$ of disconnected $\mathcal{C}(G,X)$ when $G=\mathrm{Sym}\left(6\right)$ and $X={(1,2,3)}^{G}$.

**Figure 2.**Connected component ${C}_{1}$ of disconnected $\mathcal{C}(G,X)$ when $G=\mathrm{Sym}\left(6\right)$ and $X=(1,2,3){(4,5,6)}^{G}$.

**Table 1.**Number of connected components, ${C}_{i}$ and its size of some disconnected $\mathcal{C}(G,X)$.

t | n | $|{\mathit{C}}_{\mathit{G}}\left(\mathit{t}\right)|$ | $\left|\mathit{X}\right|$ | No. of ${\mathit{C}}_{\mathit{i}}$ | Size of Each ${\mathit{C}}_{\mathit{i}}$ |
---|---|---|---|---|---|

$(1,2,3)$ | 4 | 3 | 8 | 4 | 2 |

5 | 6 | 20 | 10 | 2 | |

6 | 18 | 40 | 10 | 4 | |

$(1,2,3)(4,5,6)$ | 6 | 18 | 40 | 10 | 4 |

7 | 18 | 280 | 70 | 4 | |

8 | 36 | 1120 | 280 | 4 | |

9 | 108 | 3360 | 280 | 12 | |

$(1,2,3)(4,5,6)(7,8,9)$ | 10 | 162 | 22,400 | 2800 | 8 |

11 | 324 | 123,200 | 15,400 | 8 | |

$(1,2,3)(4,5,6)(7,8,9)(10,11,12)$ | 13 | 1944 | 3,203,200 | 200,200 | 16 |

14 | 3888 | 22,422,400 | 1,401,400 | 16 |

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## Share and Cite

**MDPI and ACS Style**

Nawawi, A.; Said Husain, S.K.; Kamel Ariffin, M.R.
Commuting Graphs, *C*(*G*, *X*) in Symmetric Groups Sym(*n*) and Its Connectivity. *Symmetry* **2019**, *11*, 1178.
https://doi.org/10.3390/sym11091178

**AMA Style**

Nawawi A, Said Husain SK, Kamel Ariffin MR.
Commuting Graphs, *C*(*G*, *X*) in Symmetric Groups Sym(*n*) and Its Connectivity. *Symmetry*. 2019; 11(9):1178.
https://doi.org/10.3390/sym11091178

**Chicago/Turabian Style**

Nawawi, Athirah, Sharifah Kartini Said Husain, and Muhammad Rezal Kamel Ariffin.
2019. "Commuting Graphs, *C*(*G*, *X*) in Symmetric Groups Sym(*n*) and Its Connectivity" *Symmetry* 11, no. 9: 1178.
https://doi.org/10.3390/sym11091178