# Some Invariants of Circulant Graphs

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Main Theorem

#### 2.1. Polynomials

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

- (1)
- ${M}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m}),x)=n{x}^{2(n-1)}$
- (2)
- ${M}_{2}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m}),x)=n{x}^{{(n-1)}^{2}}$

**Proof.**

- (1)
- $$\begin{array}{ccc}\hfill {M}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})& =& {\displaystyle \sum _{uv\u03f5E({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})}}{x}^{[{d}_{u}+{d}_{v}]},\hfill \\ & =& {\displaystyle \sum _{uv\u03f5{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})}}{x}^{[{d}_{u}+{d}_{v}]}\hfill \\ & =& |{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})|{x}^{2(n-1)}\hfill \\ & =& n{x}^{2(n-1)}\hfill \end{array}$$
- (2)
- $$\begin{array}{ccc}\hfill {M}_{2}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})& =& {\displaystyle \sum _{uv\u03f5E({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})}}{x}^{[{d}_{u}\times {d}_{v}]}\hfill \\ & =& {\displaystyle \sum _{uv\u03f5{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})}}{x}^{[{d}_{u}\times {d}_{v}]}\hfill \\ & =& |{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})|{x}^{{(n-1)}^{2}}\hfill \\ & =& n{x}^{{(n-1)}^{2}}\hfill \end{array}$$

#### 2.2. Topological Indices

**Theorem**

**3.**

- (1)
- ${M}_{1}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)=2n(n-1)$
- (2)
- ${M}_{2}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)=n{(n-1)}^{2}$
- (3)
- ${}^{m}{M}_{2}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)=\frac{n}{{(n-1)}^{2}}$
- (4)
- ${R}_{\alpha}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)=n{\{{(n-1)}^{2}\}}^{\alpha}$
- (5)
- ${R}_{\alpha}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)=\frac{n}{{(n-1)}^{\alpha}}$
- (6)
- $SDD\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)=2n$

**Proof.**

- (1)
- ${M}_{1}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)$:$$({D}_{x}+{D}_{y})f(x,y){)\left(M({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m});x,y)\right)|}_{x=y=1}=2n(n-1)$$
- (2)
- ${M}_{2}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)$:$$\left({D}_{x}{D}_{y}\right)f(x,y){)\left(M({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m});x,y)\right)|}_{x=y=1}=n{(n-1)}^{2}$$
- (3)
- ${}^{m}{M}_{2}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)$:$$\left({S}_{x}{S}_{y}\right)f(x,y){)\left(M({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m});x,y)\right)|}_{x=y=1}=\frac{n}{{(n-1)}^{2}}$$
- (4)
- ${R}_{\alpha}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)$:$$({D}_{x}^{\alpha}{D}_{y}^{\alpha})f(x,y){)\left(M({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m});x,y)\right)|}_{x=y=1}=n{(n-1)}^{2\alpha}$$
- (5)
- ${R}_{\alpha}\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)$:$$({S}_{x}^{\alpha}{S}_{y}^{\alpha})f(x,y){)\left(M({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m});x,y)\right)|}_{x=y=1}=\frac{n}{{(n-1)}^{\alpha}}$$
- (6)
- $SDD\left({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})\right)$:$$({D}_{x}{S}_{y}+{D}_{y}{S}_{x})\left(M({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m});x,y)\right){|}_{x=y=1}=2n$$

**Theorem**

**4.**

- (1)
- $P{M}_{1}\left({M}_{n}\right)=2{(n-1)}^{n}$
- (2)
- $P{M}_{2}\left({M}_{n}\right)={\{{(n-1)}^{2}\}}^{n}$
- (3)
- $HM\left({M}_{n}\right)={\left\{2(n-1)\right\}}^{2}\left(n\right)$

**Proof.**

- (1)
- $$\begin{array}{ccc}\hfill P{M}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})& =& {\displaystyle \prod _{uv\mathsf{\u03f5}E({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})}}[{d}_{u}+{d}_{v}]\hfill \\ & =& {\displaystyle \prod _{uv\mathsf{\u03f5}{E}_{1}\left({C}_{n}\right({S}_{i}}})[{d}_{u}+{d}_{v}]\hfill \\ & =& {\left\{2(n-1)\right\}}^{|{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})|}\hfill \\ & =& 2{(n-1)}^{n}\hfill \end{array}$$
- (2)
- $$\begin{array}{ccc}\hfill P{M}_{2}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})& =& {\displaystyle \prod _{uv\mathsf{\u03f5}E({C}_{n}({S}_{i}}}[{d}_{u}\times {d}_{v}]\hfill \\ & =& {\displaystyle \prod _{uv\mathsf{\u03f5}{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})}}[{d}_{u}\times {d}_{v}]\hfill \\ & =& {\{{(n-1)}^{2}\}}^{|{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})|}\hfill \\ & =& {\{{(n-1)}^{2}\}}^{n}\hfill \end{array}$$
- (3)
- $$\begin{array}{ccc}\hfill HM({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})& =& {\displaystyle \sum _{uv\mathsf{\u03f5}E({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})}}{[{d}_{u}+{d}_{v}]}^{2}\hfill \\ & =& {\displaystyle \sum _{uv\mathsf{\u03f5}{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})}}{[{d}_{u}+{d}_{v}]}^{2}\hfill \\ & =& {\left\{2(n-1)\right\}}^{2}|{E}_{1}({C}_{n}({a}_{1},{a}_{2},\dots ,{a}_{m})|\hfill \\ & =& {\left\{2(n-1)\right\}}^{2}\left(n\right)\hfill \end{array}$$

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Topological Index | $\mathit{f}(\mathit{x},\mathit{y})$ | Derivation from $\mathit{M}(\mathit{G},\mathit{x},\mathit{y})$ |
---|---|---|

First Zagreb | $x+y$ | $({D}_{x}+{D}_{y})\left(M(G;x,y)\right){|}_{x=y=1}$ |

Second Zagreb | $xy$ | $\left({D}_{x}{D}_{y}\right)\left(M(G;x,y)\right){|}_{x=y=1}$ |

${}^{m}{M}_{2}\left(G\right)$ | $\frac{1}{xy}$ | $\left({S}_{x}{D}_{y}\right)\left(M(G;x,y)\right){|}_{x=y=1}$ |

General Randić $\alpha \mathsf{\u03f5}N$ | ${\left(xy\right)}^{\alpha}$ | $({D}_{x}^{\alpha}{D}_{y}^{\alpha})\left(M(G;x,y)\right){|}_{x=y=1}$ |

General Randić $\alpha \mathsf{\u03f5}N$ | ${\frac{1}{xy}}^{\alpha}$ | $({S}_{x}^{\alpha}{S}_{y}^{\alpha})\left(M(G;x,y)\right){|}_{x=y=1}$ |

Symmetric Division Index | $\frac{{x}^{2}+{y}^{2}}{xy}$ | $({D}_{x}{S}_{y}+{D}_{y}{S}_{x})\left(M(G;x,y)\right){|}_{x=y=1}$ |

Graph | Symbol | Graph | Symbol |
---|---|---|---|

2-path graph | ${C}_{{i}_{2}}\left(1\right)$ | triangle graph | ${C}_{{i}_{3}}\left(1\right)$ |

square graph | ${C}_{{i}_{4}}\left(1\right)$ | tetrahedral graph | ${C}_{{i}_{4}}(1,2)$ |

5-cycle graph | ${C}_{{i}_{5}}\left(1\right)$ | pentatope graph | ${C}_{{i}_{5}}(1,2)$ |

6-cycle graph | ${C}_{{i}_{6}}\left(1\right)$ | octahedral graph | ${C}_{{i}_{6}}(1,2)$ |

utility graph | ${C}_{{i}_{6}}(1,3)$ | 3-prism graph | ${C}_{{i}_{6}}(2,3)$ |

6-complete graph | ${C}_{{i}_{6}}(1,2,3)$ | 7-cycle graph | ${C}_{{i}_{7}}\left(1\right)$ |

7-complete graph | ${C}_{{i}_{7}}(1,2,3)$ | 8-cycle graph | ${C}_{{i}_{8}}\left(1\right)$ |

4-antiprism graph | ${C}_{{i}_{8}}(1,2)$ | (4,4)-complete bipartite graph | ${C}_{{i}_{8}}(1,3)$ |

4-Möbius ladder graph | ${C}_{{i}_{8}}(1,4)$ | 16-cell graph | ${C}_{{i}_{8}}(1,2,3)$ |

8-complete graph | ${C}_{{i}_{8}}(1,2,3,4)$ | 9-cycle graph | ${C}_{{i}_{9}}\left(1\right)$ |

9-complete graph | ${C}_{{i}_{9}}(1,2,3,4)$ | 10-cycle graph | ${C}_{{i}_{10}}\left(1\right)$ |

5-antiprism graph | ${C}_{{i}_{10}}(1,2)$ | 5-crown graph | ${C}_{{i}_{10}}(1,3)$ |

5-Möbius ladder graph | ${C}_{{i}_{10}}(1,5)$ | 5-prism graph | ${C}_{{i}_{10}}(2,5)$ |

5-cocktail party graph | ${C}_{{i}_{10}}(1,2,3,4)$ | (5,5)-complete bipartite graph | ${C}_{{i}_{10}}(1,3,5)$ |

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**MDPI and ACS Style**

Munir, M.; Nazeer, W.; Shahzadi, Z.; Kang, S.M.
Some Invariants of Circulant Graphs. *Symmetry* **2016**, *8*, 134.
https://doi.org/10.3390/sym8110134

**AMA Style**

Munir M, Nazeer W, Shahzadi Z, Kang SM.
Some Invariants of Circulant Graphs. *Symmetry*. 2016; 8(11):134.
https://doi.org/10.3390/sym8110134

**Chicago/Turabian Style**

Munir, Mobeen, Waqas Nazeer, Zakia Shahzadi, and Shin Min Kang.
2016. "Some Invariants of Circulant Graphs" *Symmetry* 8, no. 11: 134.
https://doi.org/10.3390/sym8110134