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

Abstract

‎Let $G=(V,E)$ be a simple graph with $n\ge 3$ 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 ‎$J=\{I=(r_1, r_2,\ldots,r_k) \‎, ‎| \‎, ‎1\le r_1<r_2<\cdots<r_k\le‎ ‎n\}$ a set of all subsets of $S$ of cardinality $k$‎, ‎$1\le k\le‎ ‎n-1$‎. ‎In addition‎, ‎denote by‎ $d_{I}=d_{r_1}+d_{r_2}+\cdots+d_{r_k}$‎, ‎$1\le k\le n-1$‎, ‎$1\le‎ ‎r_1<r_2<\cdots<r_k\le n-1$‎, ‎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 ‎$S_{\alpha,k}(G)=\sum_{I\in J}d_I^{\alpha}$‎, ‎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.

Keywords

• graphs
• vertex degrees
• graph invariants