# Trees with Minimum Weighted Szeged Index Are of a Large Diameter

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## Abstract

**:**

## 1. Introduction

## 2. About the Diameter of Trees with Minimum Weighted Szeged Index

**Theorem**

**1.**

**Proof.**

**Case 1:**$\left|A\right|\ge 2$. Here, we consider first the edges ${y}_{0}{y}_{1},{y}_{1}{y}_{2},\dots ,{y}_{s-2}{y}_{s-1}$. As $\left|A\right|\ge 2$ this list is non-empty. We will introduce additional notation. By ${A}_{i}$ we will denote the set of the vertices of the component of $A-{y}_{i-1}{y}_{i}$ that contains ${y}_{i}$.

**Case 2:**$\left|A\right|=1$. In this case, the tree ${T}_{min}$ has only the vertex a in the set A and, in this case, $x=a$ and $deg\left(a\right)=1$. Then the degree of x in ${T}^{\prime}$ is 2. The contribution of the edges $av$ and $vb$ in $wSz\left({T}_{min}\right)$ is

**Corollary**

**1.**

## 3. Two Properties of Trees Having Minimum Weighted Szeged Index

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Atanasov, R.; Furtula, B.; Škrekovski, R.
Trees with Minimum Weighted Szeged Index Are of a Large Diameter. *Symmetry* **2020**, *12*, 793.
https://doi.org/10.3390/sym12050793

**AMA Style**

Atanasov R, Furtula B, Škrekovski R.
Trees with Minimum Weighted Szeged Index Are of a Large Diameter. *Symmetry*. 2020; 12(5):793.
https://doi.org/10.3390/sym12050793

**Chicago/Turabian Style**

Atanasov, Risto, Boris Furtula, and Riste Škrekovski.
2020. "Trees with Minimum Weighted Szeged Index Are of a Large Diameter" *Symmetry* 12, no. 5: 793.
https://doi.org/10.3390/sym12050793