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Sharp upper bound on the Sombor index of bipartite graphs with a given diameter

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Abstract

Let G be a connected graph. The Sombor index of a graph G is defined as \(SO(G)=\sum _{uv\in E(G)}\sqrt{d^2_{G}(u)+d^2_{G}(v)}\), where \(d_G(u)\) denotes the degree of u in G. Let \({\mathscr {B}}^d_n\) be the set of all bipartite graphs of diameter d with n vertices. In this paper, we determine the sharp upper bound on the Sombor index of \(G\in {\mathscr {B}}^d_n\). In addition, we propose an algorithm for searching the largest Sombor index among \({\mathscr {B}}^d_n\). Furthermore, the relationship between the maximal Sombor index of \(G\in {\mathscr {B}}^d_n\) and the diameter d is established. Finally, we obtain the largest, the second-largest, the third-largest and the smallest Sombor indices of bipartite graphs.

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Acknowledgements

This work is supported by the Project of Chizhou University(Grant Nos. CZ2021ZRZ03, CZ2022YJRC08, CZ2023ZRZ04), the Applied Mathematics Research Center of Chizhou University and the Hubei Provincial Natural Science Foundation and Huangshi of China (Grant Nos. 2022CFD042, 2022CFB484).

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Authors and Affiliations

  1. School of Big Data and Artificial Intelligence, Chizhou University, Chizhou, 247000, Anhui, People’s Republic of China

    Zhen Wang, Fang Gao & Duoduo Zhao

  2. School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, Hubei, People’s Republic of China

    Hechao Liu

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

    Hechao Liu

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  1. Zhen Wang
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  2. Fang Gao
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Correspondence to Hechao Liu.

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Wang, Z., Gao, F., Zhao, D. et al. Sharp upper bound on the Sombor index of bipartite graphs with a given diameter. J. Appl. Math. Comput. 70, 27–46 (2024). https://doi.org/10.1007/s12190-023-01955-8

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  • DOI: https://doi.org/10.1007/s12190-023-01955-8

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