# Extremal Graphs for Sombor Index with Given Parameters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Proposition 1.**

**Lemma 1**

**.**Let ${P}_{n}$ be the path of order n. Then for any connected graph G of order n,

**Lemma 2**

**.**Let ${S}_{n}$ be the star of order n. Then for any tree T of order n.

**Lemma 3**

**Lemma 4.**

**Lemma 5.**

**Proof.**

## 3. Connected Graphs with Given Parameters

#### 3.1. Extremal Graphs with Regard to $SO\left(G\right)$ in Terms of Order n and Chromatic Number c

**Theorem 1**

**.**Let $G\in {\mathcal{X}}_{n}^{c},q=\lfloor \frac{n}{c}\rfloor $ and $r=n-cq$. Then

**Theorem 2.**

**Proof.**

#### 3.2. Extremal Graphs with Regard to $SO\left(G\right)$ in Terms of Order n and Girth g

**Theorem 3.**

**Proof.**

#### 3.3. Extremal Graphs with Regard to ${\xi}^{d}\left(G\right)-{D}^{\prime}\left(G\right)$ in Terms of Matching Number

**Theorem 4.**

**Proof.**

## 4. Bipartite Graphs with Given Parameters

#### 4.1. Extremal Bipartite Graphs with Regard to $SO\left(G\right)$ in Terms of Matching Number $\beta $

**Theorem 5.**

**Proof.**

#### 4.2. Extremal Bipartite Graphs with Regard to $SO\left(G\right)$ in Terms of Connectivity k

**Theorem 6.**

**Proof.**

## 5. Concluding Remarks

**Problem 1.**

**Problem 2.**

**Problem 3.**

**Problem 4.**

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Gutman, I. Geometric approach to degree-based topological indices: Sombor indices. MATCH Commun. Math. Comput. Chem.
**2021**, 86, 11–16. [Google Scholar] - Réti, T.; Doslic, T.; Ali, A. On the Sombor index of graphs. Contrib. Math.
**2021**, 3, 11–18. [Google Scholar] - Liu, H.C.; You, L.H.; Tang, Z.K.; Liu, J.B. On the reduced Sombor index and its applications. MATCH Commun. Math. Comput. Chem.
**2021**, 86, 729–753. [Google Scholar] - Wang, F.X.; Wu, B. The Proof of a Conjecture on the Reduced Sombor Index. MATCH Commun. Math. Comput. Chem.
**2022**, 88, 583–591. [Google Scholar] [CrossRef] - Wang, F.X.; Wu, B. The reduced Sombor index and the exponential reduced Sombor index of a molecular tree. J. Math. Anal. Appl.
**2022**, 515, 126442. [Google Scholar] [CrossRef] - Sun, X.; Du, J. On Sombor index of trees with fixed domination number. Appl. Math. Comput.
**2022**, 421, 126946. [Google Scholar] [CrossRef] - Zhou, T.; Lin, Z.; Miao, L. The Sombor index of trees and unicyclic graphs with given matching number. arXiv
**2021**, arXiv:2103.04645. [Google Scholar] [CrossRef] - Zhou, T.; Lin, Z.; Miao, L. The Sombor index of trees and unicyclic graphs with given maximum degree. arXiv
**2021**, arXiv:2103.07947. [Google Scholar] - Das, K.C.; Gutman, I. On Sombor index of trees. Appl. Math. Comput.
**2022**, 412, 12675. [Google Scholar] [CrossRef] - Liu, H.; Gutman, I.; You, L.; Huang, Y. Sombor index:review of extremal results and bounds. J. Math. Chem.
**2022**, 60, 771–798. [Google Scholar] [CrossRef] - Aashtab, A.; Akbari, S.; Madadinia, S.; Noei, M.; Salehi, F. On the graphs with minimum Sombor index. MATCH Commun. Math. Comput. Chem.
**2022**, 88, 553–559. [Google Scholar] [CrossRef] - Liu, H.; You, L.; Huang, Y. Extremal Sombor indices of tetracyclic (chemical) graphs. MATCH Commun. Math. Comput. Chem.
**2022**, 88, 573–581. [Google Scholar] [CrossRef] - Ning, W.; Song, Y.; Wang, K. More on Sombor index of graphs. Mathematics
**2022**, 10, 301. [Google Scholar] [CrossRef] - Horoldagva, B.; Xu, C. On Sombor index of graphs. MATCH Commun. Math. Comput. Chem.
**2021**, 86, 703–713. [Google Scholar] - Das, K.C.; Cevik, A.S.; Cangul, I.N.; Shang, Y. On Sombor index. Symmetry
**2021**, 13, 140. [Google Scholar] [CrossRef] - Ghanbari, N.; Alikhani, S. Sombor index of certain graphs. Iran. J. Math. Chem.
**2021**, 12, 27–37. [Google Scholar] - Deng, H.; Tang, Z.; Wu, R. Molecular trees with extremal values of Sombor indices. Int. J. Quantum Chem.
**2021**, 121, e26622. [Google Scholar] [CrossRef] - Cruz, R.; Gutman, I.; Rada, J. Sombor index of chemical graphs. Appl. Math. Comput.
**2021**, 399, 126018. [Google Scholar] [CrossRef] - Liu, H.; Chen, H.; Xiao, Q.; Fang, X.; Tang, Z. More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons. Int. J. Quantum Chem.
**2021**, 121, 26689. [Google Scholar] [CrossRef] - Liu, H.; You, L.; Huang, Y. Ordering chemical graphs by Sombor indices and its applications. MATCH Commun. Math. Comput. Chem.
**2022**, 87, 5–22. [Google Scholar] [CrossRef] - Filipovski, S. Relations between Sombor index and some degree-based topological indices. Iran. J. Math. Chem.
**2021**, 12, 19–26. [Google Scholar] - Rada, J.; Rodriguez, J.M.; Sigarreta, J.M. General properties on Sombor indices. Discrete Appl. Math.
**2021**, 299, 87–97. [Google Scholar] [CrossRef] - Wang, F.X.; Wu, B. The k-Sombor index of trees. Asia-Pac. J. Oper. Res.
**2023**. [Google Scholar] [CrossRef] - Chen, H.; Li, W.; Wang, J. Extremal Values on the Sombor Index of Trees. MATCH Commun. Math. Comput. Chem.
**2022**, 87, 23–49. [Google Scholar] [CrossRef] - Milovanović, I.; Milovanović, E.; Matejić, M. On some mathematical properties of Sombor indices. Bull. Int. Math. Virtual Inst.
**2021**, 11, 341–353. [Google Scholar] - Xu, K.; Das, K.C. Some extremal graphs with respect to inverse degree. Discrete Appl. Math.
**2016**, 203, 171–183. [Google Scholar] [CrossRef] - Bondy, J.A.; Murty, U.S.R. Graph Theory with Applications; Macmillan Press: New York, NY, USA, 1976. [Google Scholar]
- Das, K.C.; Shang, Y. Some extremal graphs with respect to Sombor index. Mathematics
**2021**, 9, 1202. [Google Scholar] [CrossRef] - Li, S.; Song, Y. On the sum of all distances in bipartite graphs. Discrete Appl. Math.
**2014**, 169, 176–185. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Zhang, W.; Meng, J.; Wang, N.
Extremal Graphs for Sombor Index with Given Parameters. *Axioms* **2023**, *12*, 203.
https://doi.org/10.3390/axioms12020203

**AMA Style**

Zhang W, Meng J, Wang N.
Extremal Graphs for Sombor Index with Given Parameters. *Axioms*. 2023; 12(2):203.
https://doi.org/10.3390/axioms12020203

**Chicago/Turabian Style**

Zhang, Wanping, Jixiang Meng, and Na Wang.
2023. "Extremal Graphs for Sombor Index with Given Parameters" *Axioms* 12, no. 2: 203.
https://doi.org/10.3390/axioms12020203