Construction of non-regular $A_{\alpha }$-cospectral graphs from some join of graphs

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 ${\text{graphene}}_1$ and ${\text{graphene}}_2$ 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.