On the extremal cacti with minimum Sombor index

Qiaozhi Geng; Shengjie He; Rong-Xia Hao; Qiaozhi Geng; Shengjie He; Rong-Xia Hao

doi:10.3934/math.20231537

AIMS Mathematics

2023, Volume 8, Issue 12: 30059-30074. doi: 10.3934/math.20231537

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Research article

On the extremal cacti with minimum Sombor index

1.
School of Science, Tianjin University of Commerce, Tianjin 300134, China
2.
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

Received: 25 September 2023 Revised: 26 October 2023 Accepted: 01 November 2023 Published: 06 November 2023
MSC : 05C09, 05C90

Let $ H $ be a graph with edge set $ E_H $. The Sombor index and the reduced Sombor index of a graph $ H $ are defined as $ SO(H) = \sum\limits_{uv\in E_H}\sqrt{d_{H}(u)^{2}+d_{H}(v)^{2}} $ and $ SO_{red}(H) = \sum\limits_{uv\in E_H}\sqrt{(d_{H}(u)-1)^{2}+(d_{H}(v)-1)^{2}} $, respectively. Where $ d_{H}(u) $ and $ d_{H}(v) $ are the degrees of the vertices $ u $ and $ v $ in $ H $, respectively. A cactus is a connected graph in which any two cycles have at most one common vertex. Let $ \mathcal{C}(n, k) $ be the class of cacti of order $ n $ with $ k $ cycles. In this paper, the lower bound for the Sombor index of the cacti in $ \mathcal{C}(n, k) $ is obtained and the corresponding extremal cacti are characterized when $ n\geq 4k-2 $ and $ k\geq 2 $. Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach.
- cactus,
- Sombor index,
- reduced Sombor index,
- lower bound
Citation: Qiaozhi Geng, Shengjie He, Rong-Xia Hao. On the extremal cacti with minimum Sombor index[J]. AIMS Mathematics, 2023, 8(12): 30059-30074. doi: 10.3934/math.20231537

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Abstract

Let $ H $ be a graph with edge set $ E_H $. The Sombor index and the reduced Sombor index of a graph $ H $ are defined as $ SO(H) = \sum\limits_{uv\in E_H}\sqrt{d_{H}(u)^{2}+d_{H}(v)^{2}} $ and $ SO_{red}(H) = \sum\limits_{uv\in E_H}\sqrt{(d_{H}(u)-1)^{2}+(d_{H}(v)-1)^{2}} $, respectively. Where $ d_{H}(u) $ and $ d_{H}(v) $ are the degrees of the vertices $ u $ and $ v $ in $ H $, respectively. A cactus is a connected graph in which any two cycles have at most one common vertex. Let $ \mathcal{C}(n, k) $ be the class of cacti of order $ n $ with $ k $ cycles. In this paper, the lower bound for the Sombor index of the cacti in $ \mathcal{C}(n, k) $ is obtained and the corresponding extremal cacti are characterized when $ n\geq 4k-2 $ and $ k\geq 2 $. Moreover, the lower bound of the reduced Sombor index of cacti is obtained by similar approach.

References

[1]	J. A. Bondy, U. S. R. Murty, Graph theory, New York: Springer, 2008. https://doi.org/10.1007/978-1-84628-970-5
[2]	H. Wiener, Structral determination of paraffin boiling points, J. Am. Chem. Soc., 69 (1947), 17–20. https://doi.org/10.1021/ja01193a005 doi: 10.1021/ja01193a005
[3]	I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86 (2021), 11–16.
[4]	A. Alidadi, A. Parsian, H. Arianpoor, The minimum Sombor index for unicyclic graphs with fixed diameter, MATCH Commun. Math. Comput. Chem., 88 (2022), 561–572. https://doi.org/10.46793/match.88-3.561A doi: 10.46793/match.88-3.561A
[5]	T. Zhou, Z. Lin, L. Miao, The Sombor index of trees and unicyclic graphs with given maximum degree, Discrete Math. Lett., 7 (2021), 24–29. https://doi.org/10.47443/dml.2021.0035 doi: 10.47443/dml.2021.0035
[6]	K. C. Das, I. Gutman, On Sombor index of trees, Appl. Math. Comput., 412 (2022), 126575. https://doi.org/10.1016/j.amc.2021.126575 doi: 10.1016/j.amc.2021.126575
[7]	S. Li, Z. Wang, M. Zhang, On the extremal Sombor index of trees with a given diameter, Appl. Math. Comput., 416 (2022), 126731. https://doi.org/10.1016/j.amc.2021.126731 doi: 10.1016/j.amc.2021.126731
[8]	X. Sun, J. Du, On Sombor index of trees with fixed domination number, Appl. Math. Comput., 421 (2022), 126946. https://doi.org/10.1016/j.amc.2022.126946 doi: 10.1016/j.amc.2022.126946
[9]	R. Cruz, I. Gutman, J. Rada, Sombor index of chemical graphs, Appl. Math. Comput., 399 (2021), 126018. https://doi.org/10.1016/j.amc.2021.126018 doi: 10.1016/j.amc.2021.126018
[10]	H. Deng, Z. Tang, R. Wu, Molecular trees with extremal values of Sombor indices, Int. J. Quantum Chem., 121 (2021), e26622. https://doi.org/10.1002/qua.26622 doi: 10.1002/qua.26622
[11]	A. Ülker, A. Gürsoy, N. K. Gürsoy, The energy and Sombor index of graphs, MATCH. Commun. Math. Comput., 87 (2022), 51–58. https://doi.org/10.46793/match.87-1.051U doi: 10.46793/match.87-1.051U
[12]	B. Horoldagva, C. Xu, On Sombor index of graphs, MATCH Commun, MATCH Commun. Math. Comput. Chem., 86 (2021), 703–713.
[13]	H. Liu, L. You, Z. Tang, J. B. Liu, On the reduced Sombor index and its applications, MATCH Commun. Math. Comput. Chem., 86 (2021), 729–753.
[14]	R. Cruz, J. Rada, Extremal values of the Sombor index in unicyclic and bicyclic graphs, J. Math. Chem., 59 (2021), 1098–1116. https://doi.org/10.1007/s10910-021-01232-8 doi: 10.1007/s10910-021-01232-8
[15]	F. Wu, X. An, B. Wu, Sombor indices of cacti, AIMS Mathematics, 8 (2022), 1550–1565. https://doi.org/10.3934/math.2023078 doi: 10.3934/math.2023078
[16]	H. Liu, I. Gutman, L. You, Y. Huang, Sombor index: Review of extremal results and bounds, J. Math. Chem., 60 (2022), 771–798.

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