The $A_{\alpha}$- spectrum of graph product

Let $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of eigenvalues of $A_{\alpha}(G)$ together with multiplicities is called the $A_{\alpha}$-\emph{spectrum} of $G$. Let $G\square H$, $G[H]$, $G\times H$ and $G\oplus H$ be the Cartesian product, lexicographic product, directed product and strong product of graphs $G$ and $H$, respectively. In this paper, a complete characterization of the $A_{\alpha}$-spectrum of $G\square H$ for arbitrary graphs $G$ and $H$, and $G[H]$ for arbitrary graph $G$ and regular graph $H$ is given. Furthermore, $A_{\alpha}$-spectrum of the generalized lexicographic product $G[H_1,H_2,\ldots,H_n]$ for $n$-vertex graph $G$ and regular graphs $H_i$'s is considered. At last, the spectral radii of $A_{\alpha}(G\times H)$ and $A_{\alpha}(G\oplus H)$ for arbitrary graph $G$ and regular graph $H$ are given.