Extremal values of the modified Sombor index in trees

Kun Wang; Wenjie Ning; Yuheng Song; Kun Wang; Wenjie Ning; Yuheng Song

doi:10.3934/math.2025548

AIMS Mathematics

2025, Volume 10, Issue 5: 12092-12103. doi: 10.3934/math.2025548

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Extremal values of the modified Sombor index in trees

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2.
College of Science, China University of Petroleum (East China), Qingdao 266580, China
3.
School of Economics and Management, China University of Petroleum (East China), Qingdao 266580, China

Received: 10 April 2025 Revised: 06 May 2025 Accepted: 16 May 2025 Published: 26 May 2025
MSC : 05C07, 05C09, 05C92

Based on Euclidean metrics, Gutman put forward a novel vertex-degree-based topological index, named the Sombor index. Later, the modified version of it—the modified Sombor index—was introduced. For a simple undirected graph $ G $, the Sombor index and the modified Sombor index of $ G $ are defined as $ SO(G) = \sum\limits_{uv\in E(G)}\sqrt{d_G(u)^2+d_G(v)^2} $ and $ ^mSO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_G(u)^2+d_G(v)^2}} $, respectively, where $ d_G(w) $ denotes the degree of $ w $ in $ G $. Extremal values of $ SO $ have been intensively investigated. In this paper, we were concerned with the extremal values of the modified Sombor indices. First, we showed some graph transformations which can be used to compare the modified Sombor indices of two graphs. With these transformations, the first two maximum and minimum values of $ ^mSO $ among all trees of order $ n $ were determined. We also characterized the unique tree that minimizes $ ^mSO $ among all trees with a fixed number of pendant vertices. In addition, the molecular trees with the maximum and minimum modified Sombor indices were investigated.
- extremal value,
- Sombor index,
- modified Sombor index,
- tree,
- molecular tree
Citation: Kun Wang, Wenjie Ning, Yuheng Song. Extremal values of the modified Sombor index in trees[J]. AIMS Mathematics, 2025, 10(5): 12092-12103. doi: 10.3934/math.2025548

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Abstract

Based on Euclidean metrics, Gutman put forward a novel vertex-degree-based topological index, named the Sombor index. Later, the modified version of it—the modified Sombor index—was introduced. For a simple undirected graph $ G $, the Sombor index and the modified Sombor index of $ G $ are defined as $ SO(G) = \sum\limits_{uv\in E(G)}\sqrt{d_G(u)^2+d_G(v)^2} $ and $ ^mSO(G) = \sum\limits_{uv\in E(G)}\frac{1}{\sqrt{d_G(u)^2+d_G(v)^2}} $, respectively, where $ d_G(w) $ denotes the degree of $ w $ in $ G $. Extremal values of $ SO $ have been intensively investigated. In this paper, we were concerned with the extremal values of the modified Sombor indices. First, we showed some graph transformations which can be used to compare the modified Sombor indices of two graphs. With these transformations, the first two maximum and minimum values of $ ^mSO $ among all trees of order $ n $ were determined. We also characterized the unique tree that minimizes $ ^mSO $ among all trees with a fixed number of pendant vertices. In addition, the molecular trees with the maximum and minimum modified Sombor indices were investigated.

References

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