  Let $A(G)$ be the adjacency matrix and $D(G)$ be the diagonal matrix of the
vertex degrees of a simple connected graph $G$. Nikiforov defined the matrix
$A_{\alpha}(G)$ of the convex combinations of $D(G)$ and $A(G)$ as
$A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, for $0\leq \alpha\leq 1$. If $
\rho_{1}\geq \rho_{2}\geq \dots \geq \rho_{n}$ are the eigenvalues of
$A_{\alpha}(G)$ (which we call $\alpha$-adjacency eigenvalues of $G$), the $
\alpha $-adjacency energy of $G$ is defined as
$E^{A_{\alpha}}(G)=\sum_{i=1}^{n}\left|\rho_i-\frac{2\alpha m}{n}\right|$,
where $n$ is the order and $m$ is the size of $G$. We obtain the upper and
lower bounds for $E^{A_{\alpha}}(G) $ in terms of order $n$, size $m$ and
Zagreb index $Zg(G)$ associated to the structure of $G$. Further, we
characterize the extremal graphs attaining these bounds.
