  For every real $0\leq \alpha \leq 1$, Nikiforov defined the
$A_{\alpha}$-matrix of a graph $G$ as $A_{\alpha}(G)=\alpha
D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the
degree diagonal matrix of a graph $G$, respectively. The eigenvalues of
$A_{\alpha}(G)$ are called the $A_{\alpha}$-eigenvalues of $G$. Let
$S_k(A_{\alpha}(G))$ be the sum of $k$ largest $A_{\alpha}$-eigenvalues of $G$.
In this paper, we present several upper and lower bounds on
$S_k(A_{\alpha}(G))$ and characterize the extremal graphs for certain cases,
which can be regard as a common generalization of the sum of $k$ largest
eigenvalues of adjacency matrix and signless Laplacian matrix of graphs. In
addition, some graph operations on $S_k(A_{\alpha}(G))$ are presented.
