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

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