On $A_{\alpha}$-spectrum of central graph and central vertex join of two graphs
Let $G$ be a graph on $n$ vertices with adjacency matrix $A(G)$ and degree matrix $D(G)$. For every $\alpha \in [0, 1]$, $A_\alpha(G)$ is defined as $A_\alpha (G) = \alpha D(G) + (1 - \alpha)A(G)$. In this paper, we present the $A_\alpha$-characteristic polynomial and $A_\alpha$-spectrum of central graph of a regular graph. Moreover, the $A_\alpha$- characteristic polynomial of central vertex join of a regular graph and an arbitrary graph is calculated and the $A_\alpha$-spectrum of central vertex join of two regular graphs is estimated. As an application, we obtain infinitely many $A_\alpha$-cospectral graphs.