# New Bounds for Topological Indices on Trees through Generalized Methods

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. First Variable Zagreb Index

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Theorem**

**5.**

**Theorem**

**6.**

**Corollary**

**1.**

**Corollary**

**2.**

**Theorem**

**7.**

**Theorem**

**8.**

**Lemma**

**2.**

- ${y}_{j}=\Delta $ for every $1\le j\le {j}_{0}$;
- ${y}_{{j}_{0}+1}=2n-2-{j}_{0}\Delta -(n-{j}_{0}-1)=n-1-{j}_{0}(\Delta -1)$;
- ${y}_{j}=1$ for every ${j}_{0}+1<j\le n;$

- ${z}_{1}=\Delta $;
- ${z}_{j}=2$ for every $1<j\le n-\Delta $;
- ${z}_{j}=1$ for every $n-\Delta <j\le n;$

**Proof.**

**Theorem**

**9.**

**Theorem**

**10.**

**Theorem**

**11.**

**Theorem**

**12.**

**Theorem**

**13.**

**Theorem**

**14.**

**Theorem**

**15.**

**Theorem**

**16.**

**Corollary**

**3.**

## 3. Wiener Index and Its Generalizations

**Lemma**

**3.**

**Proof.**

**Theorem**

**17.**

**Proof.**

**Theorem**

**18.**

**Proof.**

**Theorem**

**19.**

**Proof.**

**Theorem**

**20.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Martínez-Pérez, Á.; Rodríguez, J.M.
New Bounds for Topological Indices on Trees through Generalized Methods. *Symmetry* **2020**, *12*, 1097.
https://doi.org/10.3390/sym12071097

**AMA Style**

Martínez-Pérez Á, Rodríguez JM.
New Bounds for Topological Indices on Trees through Generalized Methods. *Symmetry*. 2020; 12(7):1097.
https://doi.org/10.3390/sym12071097

**Chicago/Turabian Style**

Martínez-Pérez, Álvaro, and José M. Rodríguez.
2020. "New Bounds for Topological Indices on Trees through Generalized Methods" *Symmetry* 12, no. 7: 1097.
https://doi.org/10.3390/sym12071097