On the $A_{\alpha }$-characteristic polynomial of a graph

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 $A_{\alpha }$-characteristic polynomial of $G$ is defined to be $$\det (x{I}_n-A_{\alpha }(G))=\sum _j{c}_{\alpha j}(G)x^{n-j},$$ where $\det (\ast )$ denotes the determinant of $\ast$, and $I_n$ is the identity matrix of size $n$. The $A_{\alpha }$-spectrum of $G$ consists of all roots of the $A_{\alpha }$-characteristic polynomial of $G$. A graph $G$ is said to be determined by its $A_{\alpha }$-spectrum if all graphs having the same $A_{\alpha }$-spectrum as $G$ are isomorphic to $G$. In this paper, we first formulate the first four coefficients $c_{\alpha 0}(G)$,$c_{\alpha 1}(G)$,$c_{\alpha 2}(G)$and$c_{\alpha 3}(G)$ofthe$A_{\alpha}$-characteristicpolynomialof$G$\mathrm{.}Andthen,weobservethat$A_{\alpha}$-spectra are much efficient for us to distinguish graphs, by enumerating the $A_{\alpha }$-characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their $A_{\alpha }$-spectra.