# Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Lemma**

**1.**

**Theorem**

**1.**

**Theorem**

**2.**

## 3. Extended Cut Methods on Some Subsets of Vertices

**Theorem**

**3.**

**Corollary**

**1.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Corollary**

**2.**

## 4. Some Bounds on the Modified Wiener Index and the Number of Orbits

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Theorem**

**7.**

- 1.
- If T is a centered tree, then$$MW\left(G\right)\le nr(n-\frac{3r+1}{2})$$
- 2.
- If T is a bicentered tree, then$$MW\left(G\right)\le n(r-1)(n-\frac{3r}{2})+\frac{n}{2}(n-r)$$

**Proof.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The colored graph G, (

**b**) The weighted quotient graph $G/{E}_{1},$ (

**c**) The weighted quotient graph $G/{E}_{2}$.

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Nadjafi-Arani, M.J.; Mirzargar, M.; Emmert-Streib, F.; Dehmer, M.
Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications. *Symmetry* **2020**, *12*, 2027.
https://doi.org/10.3390/sym12122027

**AMA Style**

Nadjafi-Arani MJ, Mirzargar M, Emmert-Streib F, Dehmer M.
Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications. *Symmetry*. 2020; 12(12):2027.
https://doi.org/10.3390/sym12122027

**Chicago/Turabian Style**

Nadjafi-Arani, Mohammad Javad, Mahsa Mirzargar, Frank Emmert-Streib, and Matthias Dehmer.
2020. "Partition and Colored Distances in Graphs Induced to Subsets of Vertices and Some of Its Applications" *Symmetry* 12, no. 12: 2027.
https://doi.org/10.3390/sym12122027