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 $alphain [0,1]$. The $A_{alpha}$-characteristic polynomial of $G$ is defined to be $$ det(xI_n-A_{alpha}(G))=sum_jc_{alpha j}(G)x^{n-j}, $$ where $det(*)$ denotes the determinant of $*$, 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 $c_{alpha 3}(G)$ of the $A_{alpha}$-characteristic polynomial of $G$. And then, we observe that $A_{alpha}$-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.