  Cospectral graphs are a fascinating concept in graph theory, where two
non-isomorphic graphs possess identical sets of eigenvalues. In this paper, we
compute the $A_\alpha$-characteristic polynomial of neighbour and non-neighbour
splitting join, neighbour and non-neighbour shadow join, central vertex and
edge join and duplicate join of two graphs. In addition, when $\graphene_1$ and
$\graphene_2$ are regular, we compute the $A_\alpha$-spectrum of these graphs.
As an application, we construct non-regular, non-isomorphic graphs that are
$A_\alpha$-cospectral.
