The Sombor index and coindex of two-trees

Zenan Du; Lihua You; Hechao Liu; Yufei Huang; Zenan Du; Lihua You; Hechao Liu; Yufei Huang

doi:10.3934/math.2023967

AIMS Mathematics

2023, Volume 8, Issue 8: 18982-18994. doi: 10.3934/math.2023967

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The Sombor index and coindex of two-trees

1.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2.
Department of Mathematics Teaching, Guangzhou Civil Aviation College, Guangzhou 510403, China

Received: 12 April 2023 Revised: 22 May 2023 Accepted: 28 May 2023 Published: 05 June 2023
MSC : 05C09, 05C92

The Sombor index of a graph $ G $, introduced by Ivan Gutman, is defined as the sum of the weights $ \sqrt{d_G(u)^2+d_G(v)^2} $ of all edges $ uv $ of $ G $, where $ d_G(u) $ denotes the degree of vertex $ u $ in $ G $. The Sombor coindex was recently defined as $ \overline{SO}(G) = \sum_{uv\notin E(G)}\sqrt{d_G(u)^2+d_G(v)^2} $. As a new vertex-degree-based topological index, the Sombor index is important because it has been proved to predict certain physicochemical properties. Two-trees are very important structures in complex networks. In this paper, the maximum and second maximum Sombor index, the minimum and second minimum Sombor coindex of two-trees and the extremal two-trees are determined, respectively. Besides, some problems are proposed for further research.
- Sombor index,
- Sombor coindex,
- two-tree
Citation: Zenan Du, Lihua You, Hechao Liu, Yufei Huang. The Sombor index and coindex of two-trees[J]. AIMS Mathematics, 2023, 8(8): 18982-18994. doi: 10.3934/math.2023967

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Abstract

The Sombor index of a graph $ G $, introduced by Ivan Gutman, is defined as the sum of the weights $ \sqrt{d_G(u)^2+d_G(v)^2} $ of all edges $ uv $ of $ G $, where $ d_G(u) $ denotes the degree of vertex $ u $ in $ G $. The Sombor coindex was recently defined as $ \overline{SO}(G) = \sum_{uv\notin E(G)}\sqrt{d_G(u)^2+d_G(v)^2} $. As a new vertex-degree-based topological index, the Sombor index is important because it has been proved to predict certain physicochemical properties. Two-trees are very important structures in complex networks. In this paper, the maximum and second maximum Sombor index, the minimum and second minimum Sombor coindex of two-trees and the extremal two-trees are determined, respectively. Besides, some problems are proposed for further research.

References

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