# M-Polynomial and Degree-Based Topological Indices of Polyhex Nanotubes

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Results and Discussion

#### 2.1. Zigzag Polyhex Nanotubes

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### 2.2. Armchair Polyhex Nanotubes

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Topological Index | $f\left(x,y\right)$ | Derivation from $M\left(G;x,y\right)$ |

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

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

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

General Randi$\stackrel{\xb4}{\mathrm{c}}\text{}\mathsf{\alpha}\in \mathbb{N}$ | ${\left(xy\right)}^{\alpha}$ | $({D}_{x}^{\alpha}{D}_{y}^{\alpha})\left(M\left(G;x,y\right)\right){|}_{x=y=1}$ |

General Randi$\stackrel{\xb4}{\mathrm{c}\text{}}\mathsf{\alpha}\in \mathbb{N}$ | $\frac{1}{{\left(xy\right)}^{\alpha}}$ | $({S}_{x}^{\alpha}{S}_{y}^{\alpha})\left(M\left(G;x,y\right)\right){|}_{x=y=1}$ |

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

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Munir, M.; Nazeer, W.; Rafique, S.; Kang, S.M.
M-Polynomial and Degree-Based Topological Indices of Polyhex Nanotubes. *Symmetry* **2016**, *8*, 149.
https://doi.org/10.3390/sym8120149

**AMA Style**

Munir M, Nazeer W, Rafique S, Kang SM.
M-Polynomial and Degree-Based Topological Indices of Polyhex Nanotubes. *Symmetry*. 2016; 8(12):149.
https://doi.org/10.3390/sym8120149

**Chicago/Turabian Style**

Munir, Mobeen, Waqas Nazeer, Shazia Rafique, and Shin Min Kang.
2016. "M-Polynomial and Degree-Based Topological Indices of Polyhex Nanotubes" *Symmetry* 8, no. 12: 149.
https://doi.org/10.3390/sym8120149