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Vertex-degree function index for concave functions of graphs with a given clique number

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Abstract

For any connected graph \(G=(V,E)\) and any function f on the positive integers set \(\mathbb {Z}^+,\) vertex-degree function index \(H_f(G)\) is defined as the sum of \(f(d_G(v))\) over \(v\in V\), where \(d_G(v)\) is the degree of v in G. For any \(n,k\in \mathbb {Z}^+\) with \(n\ge k,\) connected graphs with n vertices and clique number k form the set \(\mathcal {W}_{n,k}\). In this paper, for any strictly concave and increasing function f on \(\mathbb {Z}^+,\) we determine the maximal and minimal values of \(H_f(G)\) over \(G\in \mathcal {W}_{n,k},\) and characterize the corresponding graphs \(G\in \mathcal {W}_{n,k}\) with the extremal values. We also get the maximum vertex-degree function index \(H_f(G)\), where f(x) is a strictly concave and decreasing function for \(x\ge 1\).

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Acknowledgements

We would like to thank the anonymous referees for their valuable comments and helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Jiaxiang Yang & Yixiang Wang

  2. Huangshi Key Laboratory of Metaverse and Virtual Simulation, School of Mathematics and Statistics, Hubei Normal University, Huangshi, 435002, People’s Republic of China

    Hechao Liu

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  1. Jiaxiang Yang
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Correspondence to Jiaxiang Yang.

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Supported by CNNSF (No. 12171410), Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai (No. 2019RS1057), the Project of Hubei Normal University (Grant No. HS2023RC057), the Graduate Innovation Project of Xiangtan University (No. XDCX2023Y105), the Hunan Provincial Innovation Foundation For Postgraduate(No. CX20230607), and the Philosophy and Social Science Research Project of Hubei Provincial Department of Education(Grant No. 23Y122).

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Yang, J., Liu, H. & Wang, Y. Vertex-degree function index for concave functions of graphs with a given clique number. J. Appl. Math. Comput. 70, 2197–2208 (2024). https://doi.org/10.1007/s12190-024-02043-1

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  • DOI: https://doi.org/10.1007/s12190-024-02043-1

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