# Topological Characterization of the Symmetrical Structure of Bismuth Tri-Iodide

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## Abstract

**:**

## 1. Introduction

## 2. Applications of Topological Indices

## 3. Methods

## 4. Bismuth Tri-Iodide

#### 4.1. Results for Bismuth Tri-Iodide Chain ($m-Bi{I}_{3}$)

**Atom-bond connectivity index.**

**The geometric arithmetic index $\mathit{GA}\left(\mathit{m}\mathbf{-}{\mathit{BiI}}_{\mathbf{3}}\right)$.**

**The general Randić index ${\mathit{R}}_{\mathit{\alpha}}\left(\mathit{m}\mathbf{-}{\mathit{BiI}}_{\mathbf{3}}\right)$.**

**First, second, and hyper Zagreb indices.**

**The fourth atom-bond connectivity index ${\mathit{ABC}}_{\mathbf{4}}\left(\mathit{m}\mathbf{-}{\mathit{BiI}}_{\mathbf{3}}\right)$.**

**The fifth geometric arithmetic index ${\mathit{GA}}_{\mathbf{5}}\left(\mathit{m}\mathbf{-}{\mathit{BiI}}_{\mathbf{3}}\right)$.**

#### 4.2. Results for Bismuth Tri-Iodide Sheet $Bi{I}_{3}(m\times n)$

**The atom-bond connectivity index $\mathit{ABC}\left({\mathit{BiI}}_{\mathbf{3}}(\mathit{m}\mathbf{\times}\mathit{n})\right)$.**

**The geometric arithmetic index $\mathit{GA}\left({\mathit{BiI}}_{\mathbf{3}}(\mathit{m}\mathbf{\times}\mathit{n})\right)$.**

**The General Randić index ${\mathit{R}}_{\mathit{\alpha}}\left({\mathit{BiI}}_{\mathbf{3}}(\mathit{m}\mathbf{\times}\mathit{n})\right)$.**

**The first, second, and hyper Zagreb indices**

**The fourth atom-bond connectivity index ${\mathit{ABC}}_{\mathbf{4}}\left({\mathit{BiI}}_{\mathbf{3}}(\mathit{m}\mathbf{\times}\mathit{n})\right)$.**

**The fifth geometric arithmetic index ${\mathit{GA}}_{\mathbf{5}}\left({\mathit{BiI}}_{\mathbf{3}}(\mathit{m}\mathbf{\times}\mathit{n})\right)$.**

## 5. Comparisons and Discussion

- For the comparison of these indices numerically for $m-Bi{I}_{3}$, we computed all indices for different values of m. Now, from Table 5, we can easily see that all indices are in increasing order as the values of m are increasing. The graphical representations of the topological indices for $m-Bi{I}_{3}$ are depicted in Figure 4, Figure 5, Figure 6 and Figure 7 for certain values of m.

- We computed all indices numerically for $Bi{I}_{3}(m\times n)$ for different values of $m,n$. We can easily see, from Table 6, that all indices are in increasing order as the values of $m,n$ are increasing. The graphical representations of the topological indices for $Bi{I}_{3}(m\times n)$ are depicted in Figure 8, Figure 9, Figure 10 and Figure 11 for certain values of $m,n$.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**The graphical representation of $ABC\left(m-Bi{I}_{3}\right)$ index in blue and that of the $GA\left(m-Bi{I}_{3}\right)$ index in green.

**Figure 5.**The graphical representation of the Randić index for $\alpha =1$ in green, that for $\alpha =-1$ in red, that for $\alpha =\frac{1}{2}$ in yellow, and that for $\alpha =-\frac{1}{2}$ in blue.

**Figure 6.**The graphical representation of the first, second, and hyper Zagreb indices in cyan, pink, and green, respectively.

**Figure 7.**The graphical representation of $AB{C}_{4}\left(m-Bi{I}_{3}\right)$ index in blue and that of the $G{A}_{5}\left(m-Bi{I}_{3}\right)$ index in brown.

**Figure 8.**The graphical representation of the $ABC\left(Bi{I}_{3}(m\times n)\right)$ index in green, and that of the $GA\left(Bi{I}_{3}(m\times n)\right)$ index in blue.

**Figure 9.**The graphical representation of the Randić index for $\alpha =1$ in red, that for $\alpha =-1$ in green, that for $\alpha =\frac{1}{2}$ in pink, and that for $\alpha =\frac{-1}{2}$ in blue.

**Figure 10.**The graphical representation of the first, second, and hyper Zagreb indices in blue, green, and yellow, respectively.

**Figure 11.**The graphical representation of $AB{C}_{4}(Bi{I}_{3}(m\times n))$ index in blue, and that of the $G{A}_{5}(Bi{I}_{3}(m\times n))$ index in brown.

$({\mathit{d}}_{\mathit{e}},{\mathit{d}}_{\mathit{f}})$ | $(1,6)$ | $(2,6)$ |
---|---|---|

Number of Edges | $4m+8$ | $20m+4$ |

$({\mathit{S}}_{\mathit{e}},{\mathit{S}}_{\mathit{f}})$ | $(6,10)$ | $(10,12)$ | $(12,12)$ |
---|---|---|---|

Number of Edges | $4m+8$ | $8m+16$ | $12m-12$ |

$({\mathit{d}}_{\mathit{e}},{\mathit{d}}_{\mathit{f}})$ | $(1,6)$ | $(2,6)$ | $(3,6)$ |
---|---|---|---|

Number of Edges | $4m+4n+4$ | $12mn+8m+8n-4$ | $6mn-6n$ |

**Table 4.**Edge partition of $Bi{I}_{3}(m\times n)$ based on the degree sum of the end vertices of each edge.

$({\mathit{S}}_{\mathit{e}},{\mathit{S}}_{\mathit{f}})$ | Number of Edges |
---|---|

$(6,10)$ | $4n+8$ |

$(6,12)$ | $4m-4$ |

$(10,12)$ | $8n+16$ |

$(12,12)$ | $16m+12n-28$ |

$(12,14)$ | $12mn-8m-12n+8$ |

$(12,18)$ | $4m-4$ |

$(14,18)$ | $6mn-4m-6n+4$ |

m | $\mathit{ABC}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{-1}$ | ${\mathit{R}}_{\frac{1}{2}}$ | ${\mathit{R}}_{-\frac{1}{2}}$ | $\mathit{GA}$ | ${\mathit{M}}_{1}$ | ${\mathit{M}}_{2}$ | $\mathit{HM}$ | ${\mathit{ABC}}_{4}$ | ${\mathit{GA}}_{5}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | $26.5$ | 288 | 4 | $49.5$ | $8.5$ | $45.5$ | 276 | 360 | 2158 | $86.7$ | $181.4$ |

2 | $34.8$ | 552 | $6.2$ | $102.8$ | $15.9$ | $58.9$ | 464 | 624 | 3632 | $174.6$ | $358.5$ |

3 | $52.3$ | 816 | $8.3$ | $145.2$ | $22.8$ | $98.7$ | 652 | 888 | 5106 | $536.2$ | $536.4$ |

4 | $\mathrm{85.4.3}$ | 1080 | $11.4$ | $192.5$ | $30.5$ | $142.5$ | 840 | 1152 | 6580 | $704.2$ | $704.3$ |

$[\mathit{m},\mathit{n}]$ | $\mathit{ABC}$ | ${\mathit{R}}_{1}$ | ${\mathit{R}}_{-1}$ | ${\mathit{R}}_{\frac{1}{2}}$ | ${\mathit{R}}_{-\frac{1}{2}}$ | $\mathit{GA}$ | ${\mathit{M}}_{1}$ | ${\mathit{M}}_{2}$ | $\mathit{HM}$ | ${\mathit{ABC}}_{4}$ | ${\mathit{GA}}_{5}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

$[1,1]$ | $14.2$ | 360 | $3.4$ | $7.6$ | $81.2$ | $13.2$ | 276 | 360 | 2124 | $11.3$ | $9.8$ |

$[2,2]$ | $43.3$ | 744 | $8.6$ | $23.2$ | $234.9$ | $43.3$ | 585 | 1248 | 6816 | $50.8$ | $43.6$ |

$[3,3]$ | $96.5$ | 1128 | $19.2$ | $45.8$ | $565.3$ | $96.5$ | 1736 | 2640 | 14016 | $100.3$ | $98.5$ |

$[4,4]$ | $164.3$ | 1412 | $31.4$ | $64.3$ | $965.4$ | $164.8$ | 2918 | 4536 | 23724 | $160.6$ | $157.3$ |

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**MDPI and ACS Style**

Imran, M.; Ali, M.A.; Ahmad, S.; Siddiqui, M.K.; Baig, A.Q.
Topological Characterization of the Symmetrical Structure of Bismuth Tri-Iodide. *Symmetry* **2018**, *10*, 201.
https://doi.org/10.3390/sym10060201

**AMA Style**

Imran M, Ali MA, Ahmad S, Siddiqui MK, Baig AQ.
Topological Characterization of the Symmetrical Structure of Bismuth Tri-Iodide. *Symmetry*. 2018; 10(6):201.
https://doi.org/10.3390/sym10060201

**Chicago/Turabian Style**

Imran, Muhammad, Muhammad Arfan Ali, Sarfraz Ahmad, Muhammad Kamran Siddiqui, and Abdul Qudair Baig.
2018. "Topological Characterization of the Symmetrical Structure of Bismuth Tri-Iodide" *Symmetry* 10, no. 6: 201.
https://doi.org/10.3390/sym10060201