Some remarks on the sum of powers of the degrees of graphs

‎Let $G=(V,E)$ be a simple graph with $nge 3$ vertices‎, ‎$m$ edges‎ ‎and vertex degree sequence $Delta=d_1 ge d_2 ge cdots ge‎ ‎d_n=delta>0$‎. ‎Denote by $S={1, 2,ldots,n}$ an index set and by‎ ‎$J={I=(r_1, r_2,ldots,r_k) ‎, ‎| ‎, ‎1le r_1<r_2<cdots<r_kle‎ ‎n}$ a set of all subsets of $S$ of cardinality $k$‎, ‎$1le kle‎ ‎n-1$‎. ‎In addition‎, ‎denote by‎ $d_{I}=d_{r_1}+d_{r_2}+cdots+d_{r_k}$‎, ‎$1le kle n-1$‎, ‎$1le‎ ‎r_1<r_2<cdots<r_kle n-1$‎, ‎the sum of $k$ arbitrary vertex‎ ‎degrees‎, ‎where $Delta_{I}=d_{1}+d_{2}+cdots+d_{k}$ and‎ ‎$delta_{I}=d_{n-k+1}+d_{n-k+2}+cdots+d_{n}$‎. ‎We consider the following graph invariant‎ ‎$S_{alpha,k}(G)=sum_{Iin J}d_I^{alpha}$‎, ‎where $alpha$ is an‎ ‎arbitrary real number‎, ‎and establish its bounds‎. ‎A number of known bounds for various topological indices are obtained as special cases‎.