On the sum of powers of the $A_{\alpha }$-eigenvalues of graphs

Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. For any real number $\alpha \in [0,1]$, Nikiforov recently defined the $A_{\alpha }$-matrix of $G$ as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$. The graph invariant $S_{\alpha }^p(G)$ is the sum of the $p$-th power of the $A_{\alpha }$-eigenvalues of $G$ for $\frac{1}{2}<\alpha <1$, which has a close relation to the $\alpha$-Estrada index. In this paper, we establish some bounds on $S_{\alpha }^p(G)$ and characterize the extremal graphs. In particular, we present some bounds on $S_{\alpha }^p(G)$ in terms of the degree sequences, order and size of $G$ by using majorization techniques. Moreover, we give lower and upper bounds for $S_{\alpha }^p(G)$ of a bipartite graph and characterize the extremal graphs.