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Proof of an open problem on the Sombor index

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Abstract

The Sombor index is one of the geometry-based descriptors, which was defined as

$$\begin{aligned} SO(G)=\sum _{uv\in E(G)}\sqrt{d^{2}_{u}+d^{2}_{v}}, \end{aligned}$$

where \(d_{u}\) (resp. \(d_{v}\)) denotes the degree of vertex u (resp. v) in G. In this note, we determine the graphs among the set of graphs with vertex connectivity (resp. edge connectivity) at most k having the maximum and minimum Sombor indices, which solves an open problem on the Sombor index proposed by Hayat and Rehman [On Sombor index of graphs with a given number of cut-vertices, MATCH Commun. Math. Comput. Chem. 89 (2023) 437–450]. For some conclusions of the above paper, we first give some counterexamples, then provide another simple proof about the minimum Sombor indices of graphs with n vertices, k cut vertices and at least one cycle.

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Acknowledgements

This research is partially supported by the National Natural Science Foundation of China (Grant No. 11971180), the Guangdong Provincial Natural Science Foundation (Grant No. 2019A1515012052), the Characteristic Innovation Project of General Colleges and Universities in Guangdong Province (Grant No. 2022KTSCX225) and the Guangdong Education and Scientific Research Project (Grant No. 2021GXJK159).

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  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Hechao Liu

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Liu, H. Proof of an open problem on the Sombor index. J. Appl. Math. Comput. 69, 2465–2471 (2023). https://doi.org/10.1007/s12190-023-01843-1

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