Graphs determined by their $A_{\alpha }$-spectra

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. 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 are called the \emph{$A_{\alpha }$-spectrum} of $G$. A graph $G$ is said to be \emph{determined by its $A_{\alpha }$-spectrum} if all graphs having the same $A_{\alpha }$-spectrum as $G$ are isomorphic to $G$. We first prove that some graphs are determined by its $A_{\alpha }$-spectrum for $0\le \alpha <1$, including the complete graph $K_m$, the star $K_{1,n-1}$, the path $P_n$, the union of cycles and the complement of the union of cycles, the union of $K_2$ and $K_1$ and the complement of the union of $K_2$ and $K_1$, and the complement of $P_n$. Setting $\alpha =0$ or $\frac{1}{2}$, those graphs are determined by $A$- or $Q$-spectra. Secondly, when $G$ is regular, we show that $G$ is determined by its $A_{\alpha }$-spectrum if and only if the join $G\vee K_m$isdeterminedbyits$A_{\alpha}$-spectrumfor$\frac{1}{2}<\alpha<1$. Furthermore, we also show that the join $K_m\vee P_n$ is determined by its $A_{\alpha }$-spectrum for $\frac{1}{2}<\alpha <1$. In the end, we pose some related open problems for future study.