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

Let 𝐺 = (𝑉, 𝐸) be a simple graph with 𝑛 ≥ 3 vertices, m edges and vertex degree sequence ∆ = 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑𝑛 = δ > 0. denote by s = {1, 2, . . . , n} an index set and by 𝑗 = {i =(𝑟1, 𝑟2, . . . , 𝑟𝑘)| 1 ≤ 𝑟1 < 𝑟2 < · · · <𝑟𝑘 ≤ 𝑛} a set of all subsets of s of cardinality 𝑘, 1 ≤ 𝑘 ≤ 𝑛 − 1. in addition, denote by 𝑑i = 𝑑𝑟1 + 𝑑𝑟2 + · · · + 𝑑r𝑘 , 1 ≤ 𝑘 ≤ 𝑛 − 1, 1 ≤ 𝑟1 < 𝑟2 < · · · <𝑟𝑘 ≤ 𝑛 − 1, the sum of 𝑘 arbitrary vertex degrees, where ∆i = 𝐼1 + 𝐼2 + · · · + 𝐼𝑘 and δi = 𝐼𝑛−𝑘+1 + 𝐼𝑛−𝑘+2 + · · · + 𝑑𝑛. we consider the following graph invariant 𝑆𝑎,𝑘(𝐺) = ∑𝐼∈𝑗 𝐼 𝑎 𝐼 , where α is an arbitrary real number, and establish its bounds. a number of known bounds for various topological indices are obtained as special cases.