On the sum of the largest $A_{\alpha }$-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 defined the $A_{\alpha }$-matrix of a graph $G$ as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$. Let $S_k(A_{\alpha }(G))$ be the sum of the $k$ largest eigenvalues of $A_{\alpha }(G)$. In this paper, some bounds on $S_k(A_{\alpha }(G))$ are obtained, which not only extends the results of the sum of the $k$ largest eigenvalues of the adjacency matrix and signless Laplacian matrix, but it also gives new bounds on graph energy.