# On Topological Indices of Certain Families of Nanostar Dendrimers

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## Abstract

**:**

## 1. Introduction and Preliminary Results

## 2. Results and Discussion

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Proof.**

**Theorem 6.**

**Proof.**

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Sample Availability:**Not available.

**Figure 3.**POPAM dendrimer of generations ${G}_{n}$ with two growth stages, $PO{D}_{2}\left[2\right]$.

**Table 1.**Edge partition of PAMAM dendrimer, $P{D}_{1}\left[n\right]$ based on degree sum of neighbors of end vertices of each edge.

$\left({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}}\right)$ Where $\mathit{u}\mathit{v}\in \mathit{E}\left(\mathit{G}\right)$ | Number of Edges |
---|---|

$\left(\mathbf{2},\mathbf{3}\right)$ | $3\times {2}^{n}$ |

$\left(\mathbf{3},\mathbf{4}\right)$ | $3\times {2}^{n}$ |

$\left(\mathbf{3},\mathbf{5}\right)$ | $6\times {2}^{n}-3$ |

$\left(\mathbf{4},\mathbf{5}\right)$ | $9\times {2}^{n}-6$ |

$\left(\mathbf{5},\mathbf{5}\right)$ | $18\times {2}^{n}-9$ |

$\left(\mathbf{5},\mathbf{6}\right)$ | $9\times {2}^{n}-6$ |

**Table 2.**Edge partition of tetrathiafulvalene dendrimer $T{D}_{2}\left[n\right]$ based on degree sum of neighbors of end vertices of each edge.

$\left({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}}\right)$ Where $\mathit{u}\mathit{v}\in \mathit{E}\left(\mathit{G}\right)$ | Number of Edges |
---|---|

$\left(\mathbf{2},\mathbf{4}\right)$ | ${2}^{n+2}$ |

$\left(\mathbf{3},\mathbf{6}\right)$ | ${2}^{n+2}-4$ |

$\left(\mathbf{4},\mathbf{6}\right)$ | ${2}^{n+2}$ |

$\left(\mathbf{5},\mathbf{5}\right)$ | $7\times {2}^{n+2}-16$ |

$\left(\mathbf{5},\mathbf{6}\right)$ | $11\times {2}^{n+2}-24$ |

$\left(\mathbf{5},\mathbf{7}\right)$ | $3\times {2}^{n+2}-8$ |

$\left(\mathbf{6},\mathbf{6}\right)$ | ${2}^{n+2}-4$ |

$\left(\mathbf{6},\mathbf{7}\right)$ | $8\times {2}^{n+2}-24$ |

$\left(\mathbf{7},\mathbf{7}\right)$ | $2\times {2}^{n+2}-5$ |

**Table 3.**Edge partition of POPAM dendrimer, $PO{D}_{2}\left[n\right]$ based on degree sum of neighbors of end vertices of each edge.

$\left({\mathit{S}}_{\mathit{u}},{\mathit{S}}_{\mathit{v}}\right)$ Where $\mathit{u}\mathit{v}\in \mathit{E}\left(\mathit{G}\right)$ | Number of Edges |
---|---|

$\left(\mathbf{2},\mathbf{3}\right)$ | ${2}^{n+2}$ |

$\left(\mathbf{3},\mathbf{4}\right)$ | ${2}^{n+2}$ |

$\left(\mathbf{4},\mathbf{4}\right)$ | $1$ |

$\left(\mathbf{4},\mathbf{5}\right)$ | $3\times {2}^{n+2}-6$ |

$\left(\mathbf{5},\mathbf{6}\right)$ | $3\times {2}^{n+2}-6$ |

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**MDPI and ACS Style**

Husin, M.N.; Hasni, R.; Arif, N.E.; Imran, M.
On Topological Indices of Certain Families of Nanostar Dendrimers. *Molecules* **2016**, *21*, 821.
https://doi.org/10.3390/molecules21070821

**AMA Style**

Husin MN, Hasni R, Arif NE, Imran M.
On Topological Indices of Certain Families of Nanostar Dendrimers. *Molecules*. 2016; 21(7):821.
https://doi.org/10.3390/molecules21070821

**Chicago/Turabian Style**

Husin, Mohamad Nazri, Roslan Hasni, Nabeel Ezzulddin Arif, and Muhammad Imran.
2016. "On Topological Indices of Certain Families of Nanostar Dendrimers" *Molecules* 21, no. 7: 821.
https://doi.org/10.3390/molecules21070821