# On Eccentricity-Based Topological Indices and Polynomials of Phosphorus-Containing Dendrimers

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## Abstract

**:**

## 1. Introduction

## 2. Definitions and Notations

## 3. The Eccentricity-Based Indices and Polynomials for the Molecular Graph

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Adronov, A.; Frechet, J.M.J. Light-harvesting dendrimers. Chem. Commun.
**2000**, 33, 1701–1710. [Google Scholar] [CrossRef] - Naka, K.; Tanaka, Y.; Chujo, Y. Effect of anionic starburst dendrimers on the crystallization of CaCO
_{3}in aqueous solution, Size control of spherical vaterite particles. Langmuir**2002**, 18, 3655–3658. [Google Scholar] [CrossRef] - Suresh, R.; Singh, C.; Rewar, P. Dendrimers as carriers and its application in therapy. Int. J. Anal. Pharm. Biomed. Sci.
**2015**, 4, 15–23. [Google Scholar] - Wiener, H. Structural determination of paraffin boiling points. J. Am. Chem. Soc.
**1947**, 69, 17–20. [Google Scholar] [CrossRef] [PubMed] - Sharma, V.; Goswami, R.; Madan, A.K. Eccentric-connectivity index: A novel highly discriminating topological descriptor for structure–property and structure–activity studies. J. Chem. Inf. Comput. Sci.
**1997**, 37, 273–282. [Google Scholar] [CrossRef] - Dureja, H.; Madan, A.K. Topochemical models for prediction of cyclin-dependent kinase 2 inhibitory activity of indole-2-ones. J. Mol. Model.
**2005**, 11, 525–531. [Google Scholar] [CrossRef] [PubMed] - Ilic, A.; Gutman, I. Eccentric-connectivity index of chemical trees. MATCH Commun. Math. Comput. Chem.
**2011**, 65, 731–744. [Google Scholar] - Kumar, V.; Madan, A.K. Application of graph theory: Prediction of cytosolic phospholipase A(2) inhibitory activity of propan-2-ones. J. Math. Chem.
**2006**, 39, 511–521. [Google Scholar] [CrossRef] - Zhou, B. On eccentric-connectivity index. MATCH Commun. Math. Comput. Chem.
**2010**, 63, 181–198. [Google Scholar] - Ashrafi, A.R.; Ghorbani, M.; Hossein-Zadeh, M.A. The eccentric-connectivity polynomial of some graph operations. Serdica J. Comput.
**2011**, 5, 101–116. [Google Scholar] - Ghorbani, M.; Hosseinzadeh, M.A. A new version of Zagreb indices. Filomat
**2012**, 26, 93–100. [Google Scholar] [CrossRef] - Gupta, S.; Singh, M.; Madan, A. K Connective eccentricity index: A novel topological descriptor for predicting biological activity. J. Mol. Graph. Model.
**2000**, 18, 18–25. [Google Scholar] [CrossRef] - De, N. Relationship between augmented eccentric-connectivity index and some other graph invariants. Int. J. Adv. Math.
**2013**, 1, 26–32. [Google Scholar] [CrossRef] - Doślić, T.; Saheli, M. Augmented eccentric-connectivity index. Miskolc Math. Notes
**2011**, 12, 149–157. [Google Scholar] - Alaeiyan, M.; Asadpour, J.; Mojarad, R. A numerical method for MEC polynomial and MEC index of one-pentagonal carbon nanocones. Fuller. Nanotub. Carbon Nanostruct.
**2013**, 21, 825–835. [Google Scholar] [CrossRef] - Aslam, A.; Jamil, M.K.; Gao, W.; Nazeer, W. Topological aspects of some dendrimer structures. Nanotechnol. Rev.
**2018**, 7, 123–129. [Google Scholar] [CrossRef] - Aslam, A.; Bashir, Y.; Ahmad, S.; Gao, W. On Topological Indices of Certain Dendrimer Structures. Z. Naturforschung A
**2017**, 72, 559–566. [Google Scholar] [CrossRef] - Bashir, Y.; Aslam, A.; Kamran, M.; Qureshi, M.I.; Jahangir, A.; Rafiq, M.; Bibi, N.; Muhammad, N. On forgotten topological indices of some dendrimers structure. Molecules
**2017**, 22, 867. [Google Scholar] [CrossRef] [PubMed] - Soleimania, N.; Bahnamirib, S.B.; Nikmehr, M.J. Study of dendrimers by topological indices. ACTA CHEMICA IASI
**2017**, 25, 145–162. [Google Scholar] [CrossRef] - Wu, H.; Zhao, B.; Gao, W. Distance indices calculating for two classes of dendrimer. Geol. Ecol. Landsc.
**2017**, 1, 133–142. [Google Scholar] [CrossRef] - Yang, J.; Xia, F. The eccentric connectivity index of dendrimers. Int. J. Contemp. Math. Sci.
**2010**, 5, 2231–2236. [Google Scholar] - Badetti, E.; Lloveras, V.; Muñoz-Gómez, J.L.; Sebastián, R.M.; Camimade, A.M.; Majoral, J.P.; Veciana, J.; Vidal-Gancedo, J. Radical dendrimers: A family of five generations of phosphorus dendrimers functionalized with TEMPO radicals. Macromolecules
**2014**, 47, 7717–7724. [Google Scholar] [CrossRef]

**Table 1.**Sets A and B with their degrees, ${S}_{u}$, $M\left(u\right)$, eccentricities, and frequencies.

Representative | Degree | ${\mathit{S}}_{\mathit{u}}$ | $\mathit{M}\left(\mathit{u}\right)$ | Eccentricity | Frequency |
---|---|---|---|---|---|

${\alpha}_{1}$ | 2 | 8 | 16 | $9n+15$ | 3 |

${\alpha}_{2}$ | 4 | 8 | 16 | $9n+14$ | 3 |

${\beta}_{1}$ | 2 | 7 | 12 | $9n+15$ | $3\times {2}^{n+1}$ |

${\beta}_{2}$ | 3 | 6 | 8 | $9n+16$ | $3\times {2}^{n+1}$ |

${\beta}_{3}$ | 2 | 5 | 6 | $9n+17$ | $3\times {2}^{n+2}$ |

${\beta}_{4}$ | 2 | 5 | 6 | $9n+18$ | $3\times {2}^{n+2}$ |

${\beta}_{5}$ | 3 | 6 | 8 | $9n+19$ | $3\times {2}^{n+1}$ |

${\beta}_{6}$ | 2 | 5 | 6 | $9n+20$ | $3\times {2}^{n+1}$ |

${\beta}_{7}$ | 2 | 5 | 6 | $9n+21$ | $3\times {2}^{n+1}$ |

${\beta}_{8}$ | 3 | 6 | 8 | $9n+22$ | $3\times {2}^{n+1}$ |

${\beta}_{9}$ | 2 | 7 | 12 | $9n+23$ | $3\times {2}^{n+2}$ |

${\beta}_{10}$ | 4 | 7 | 6 | $9n+24$ | $3\times {2}^{n+2}$ |

${\beta}_{11}$ | 1 | 4 | 4 | $9n+25$ | $3\times {2}^{n+3}$ |

${\beta}_{12}$ | 3 | 9 | 16 | $9n+25$ | $3\times {2}^{n+1}$ |

${\beta}_{13}$ | 1 | 3 | 3 | $9n+26$ | $3\times {2}^{n+1}$ |

Representative | Degree | ${\mathit{S}}_{\mathit{u}}$ | $\mathit{M}\left(\mathit{u}\right)$ | Eccentricity | Frequency |
---|---|---|---|---|---|

${a}_{i}$ | 2 | 7 | 12 | $9n+9i+6=\gamma +6$ | $3\times {2}^{i}$ |

${b}_{i}$ | 3 | 6 | 8 | $\gamma +7$ | $3\times {2}^{i}$ |

${c}_{i}$ | 2 | 5 | 6 | $\gamma +8$ | $3\times {2}^{i+1}$ |

${d}_{i}$ | 2 | 5 | 6 | $\gamma +9$ | $3\times {2}^{i+1}$ |

${e}_{i}$ | 3 | 6 | 8 | $\gamma +10$ | $3\times {2}^{i}$ |

${f}_{i}$ | 2 | 5 | 6 | $\gamma +11$ | $3\times {2}^{i}$ |

${g}_{i}$ | 2 | 5 | 6 | $\gamma +12$ | $3\times {2}^{i}$ |

${h}_{i}$ | 3 | 7 | 8 | $\gamma +13$ | $3\times {2}^{i}$ |

${j}_{i}$ | 1 | 3 | 3 | $\gamma +14$ | $3\times {2}^{i}$ |

${k}_{i}$ | 4 | 8 | 12 | $\gamma +14$ | $3\times {2}^{i}$ |

${l}_{i}$ | 1 | 4 | 4 | $\gamma +15$ | $3\times {2}^{i}$ |

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**MDPI and ACS Style**

Kang, S.M.; Iqbal, Z.; Ishaq, M.; Sarfraz, R.; Aslam, A.; Nazeer, W.
On Eccentricity-Based Topological Indices and Polynomials of Phosphorus-Containing Dendrimers. *Symmetry* **2018**, *10*, 237.
https://doi.org/10.3390/sym10070237

**AMA Style**

Kang SM, Iqbal Z, Ishaq M, Sarfraz R, Aslam A, Nazeer W.
On Eccentricity-Based Topological Indices and Polynomials of Phosphorus-Containing Dendrimers. *Symmetry*. 2018; 10(7):237.
https://doi.org/10.3390/sym10070237

**Chicago/Turabian Style**

Kang, Shin Min, Zahid Iqbal, Muhammad Ishaq, Rabia Sarfraz, Adnan Aslam, and Waqas Nazeer.
2018. "On Eccentricity-Based Topological Indices and Polynomials of Phosphorus-Containing Dendrimers" *Symmetry* 10, no. 7: 237.
https://doi.org/10.3390/sym10070237