On the characteristic polynomial of the $A_{\alpha }$-matrix for some operations of graphs

Let G be a graph of order $n$ with adjacency matrix $A(G)$ and diagonal matrix of degree $D(G)$. For every $\alpha \in [0,1]$, Nikiforov \cite{VN17} defined the matrix $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$. In this paper we present the $A_{\alpha }(G)$-characteristic polynomial when $G$ is obtained by coalescing two graphs, and if $G$ is a semi-regular bipartite graph we obtain the $A_{\alpha }$-characteristic polynomial of the line graph associated to $G$. Moreover, if $G$ is a regular graph we exhibit the $A_{\alpha }$-characteristic polynomial for the graphs obtained from some operations.