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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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