# Some Computational Aspects of Boron Triangular Nanotubes

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

## 2. Results

#### 2.1. M-Polynomials

**Theorem**

**1.**

**Proof.**

#### 2.2. Topological Indices

**Proposition**

**1.**

- ${M}_{1}\left(BN{T}_{t}\left[m,n\right]\right)=54mn-60m.$
- ${M}_{2}\left(BN{T}_{t}\left[m,n\right]\right)=162mn-240m.$
- ${}^{m}M_{2}\left(BN{T}_{t}\left[m,n\right]\right)=\frac{5}{48}m+\frac{1}{8}mn.$
- ${R}_{a}\left(BN{T}_{t}\left[m,n\right]\right)=\frac{3}{2}m\left(3n{\left(36\right)}^{\alpha}+2{\left(16\right)}^{\alpha}+4{\left(24\right)}^{\alpha}-8{\left(36\right)}^{\alpha}\right).$
- $R{R}_{a}\left(BN{T}_{t}\left[m,n\right]\right)=\frac{3m}{{\left(16\right)}^{\alpha}}+\frac{m}{{6}^{\alpha -1}{4}^{\alpha}}+\frac{3m(3n-8)}{{\left(36\right)}^{\alpha}}.$
- $SDD\left(BN{T}_{t}\left[m,n\right]\right)=9mn-5m.$

**Proof.**

- ${M}_{1}\left(BN{T}_{t}\left[m,n\right]\right)\text{}={({D}_{x}+{D}_{y})\left(M\left(G;x,y\right)\right)|}_{x=y=1}=54mn-60m.$
- ${M}_{2}\left(BN{T}_{t}[m,n]\right)={{D}_{x}{D}_{y}\left(M(G;x,y)\right)|}_{x=y=1}=162mn-240m.$
- ${}^{m}M_{2}\left(BN{T}_{t}\left[m,n\right]\right)={{S}_{x}{S}_{y}\left(M\left(G;x,y\right)\right)|}_{x=y=1}=\frac{5}{48}m+\frac{1}{8}mn.$
- ${R}_{a}\left(BN{T}_{t}\left[m,n\right]\right)={{D}_{x}^{\alpha}{D}_{y}^{\alpha}\left(M(G;x,y)\right)|}_{x=y=1}=\frac{3}{2}m\left(3n{\left(36\right)}^{\alpha}+2{\left(16\right)}^{\alpha}+4{\left(24\right)}^{\alpha}-8{\left(36\right)}^{\alpha}\right).$
- $R{R}_{a}\left(BN{T}_{t}\left[m,n\right]\right)={{S}_{x}^{\alpha}{S}_{y}^{\alpha}\left(M(G;x,y)\right)|}_{x=y=1}=\frac{3m}{{\left(16\right)}^{\alpha}}+\frac{m}{{6}^{\alpha -1}{4}^{\alpha}}+\frac{3m(3n-8)}{{\left(36\right)}^{\alpha}}.$
- $SDD\left(BN{T}_{t}\left[m,n\right]\right)={\left({D}_{x}{S}_{y}+{S}_{x}{D}_{y}\right)\left(M(G;x,y)\right)|}_{x=y=1}=9mn-5m.$

**Proposition**

**2.**

- $H\left(BN{T}_{t}\text{}\left[m,n\right]\text{}\right)=-\frac{1}{20}m+\frac{3}{4}mn.$
- $I\left(BN{T}_{t}[m,n]\right)=-\frac{78}{5}m+\frac{27}{2}mn.$
- $A\left(BN{T}_{t}\left[m,n\right]\right)=-\frac{383606}{1125}m+\frac{26244}{125}mn.$

**Proof.**

- $H\left(BN{T}_{t}\left[m,n\right]\text{}\right)={2{S}_{x}J\left(M(G;x,y)\right)|}_{x=1}=-\frac{1}{20}m+\frac{3}{4}mn.$
- $I\left(BN{T}_{t}\left[m,n\right]\right)={{S}_{x}J{D}_{x}{D}_{y}\left(M(G;x,y)\right)|}_{x=1}=-\frac{78}{5}m+\frac{27}{2}mn.$
- $A\left(BN{T}_{t}\left[m,n\right]\right)={{{S}_{x}}^{3}{Q}_{-2}J{{D}_{x}}^{3}{{D}_{y}}^{3}\left(M(G;x,y)\right)|}_{x=1}=-\frac{383606}{1125}m+\frac{26244}{125}mn.$

## 3. Conclusions and Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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

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

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

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

Randić index | $\left({D}_{x}^{\alpha}{D}_{y}^{\alpha}\right){(M(G;x,y))}_{x=y=1}$ |

Inverse Randić index | $\left({S}_{x}^{\alpha}{S}_{y}^{\alpha}\right){(M(G;x,y))}_{x=y=1}$ |

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

Harmonic Index | $2{S}_{x}J{(M(G;x,y))}_{x=1}$ |

Inverse sum Index | ${S}_{x}J{D}_{x}{D}_{y}{(M(G;x,y))}_{x=1}$ |

Augmented Zagreb Index | ${{S}_{x}}^{3}{Q}_{-2}J{{D}_{x}}^{3}{{D}_{y}}^{3}{(M(G;x,y))}_{x=1}$ |

$\left({d}_{u},\text{}{d}_{v}\right)$ | $\left(4,\text{}4\right)$ | $\left(4,\text{}6\right)$ | $\left(6,\text{}6\right)$ |

Number of edges | $3m$ | $6m$ | $\frac{m}{2}(9n-24)$ |

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Munir, M.; Nazeer, W.; Rafique, S.; Nizami, A.R.; Kang, S.M.
Some Computational Aspects of Boron Triangular Nanotubes. *Symmetry* **2017**, *9*, 6.
https://doi.org/10.3390/sym9010006

**AMA Style**

Munir M, Nazeer W, Rafique S, Nizami AR, Kang SM.
Some Computational Aspects of Boron Triangular Nanotubes. *Symmetry*. 2017; 9(1):6.
https://doi.org/10.3390/sym9010006

**Chicago/Turabian Style**

Munir, Mobeen, Waqas Nazeer, Shazia Rafique, Abdul Rauf Nizami, and Shin Min Kang.
2017. "Some Computational Aspects of Boron Triangular Nanotubes" *Symmetry* 9, no. 1: 6.
https://doi.org/10.3390/sym9010006