On the sum of the powers of $ A_\alpha $ eigenvalues of graphs and $ A_\alpha $-energy like invariant

For a connected simple graph $ G $ with $ A_{\alpha} $ eigenvalues $ \rho_{1}\geq\rho_{2}\geq\dots\geq\rho_{n} $ and a real number $\beta $, let $ S_{\beta}^{\alpha}(G) =\sum\limits_{i=1}^{n}\rho_{i}^{\beta}$ be the sum of the $ \beta^{th} $ powers of the $ A_{\alpha} $ eigenvalues of graph $ G $. In this paper, we obtain various bounds for the graph invariant $ S_{\beta}^{\alpha}(G) $ in terms of different graph parameters. As a consequence, we obtain the bounds for the quantity $ IE^{A_{\alpha}}(G)= S_{\frac{1}{2}}^{\alpha}(G),$ the $ A_{\alpha} $ energy-like invariant of the graph $ G .$