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Let G be a simple, connected graph and let A(G) be the adjacency matrix of G. If D(G) is the diagonal matrix of the vertex degrees of G, then for every real α ∈ [ 0 , 1 ] , the matrix A α (G) is defined as A α (G) = α D (G) + (1 - α) A (G). The eigenvalues of the matrix A α (G) form the A α -spectrum of G. Let G 1 ∨ ˙ G 2 , G 1 ∨ ̲ G 2 , G 1 ⟨ v ⟩ G 2 and G 1 ⟨ e ⟩ G 2 denote the subdivision-vertex join, subdivision-edge join, R-vertex join and R-edge join of two graphs G 1 and G 2 , respectively. In this paper, we compute the A α -spectra of G 1 ∨ ˙ G 2 , G 1 ∨ ̲ G 2 , G 1 ⟨ v ⟩ G 2 and G 1 ⟨ e ⟩ G 2 for a regular graph G 1 and an arbitrary graph G 2 in terms of their A α -eigenvalues. As an application of these results, we construct infinitely many pairs of A α -cospectral graphs. [ABSTRACT FROM AUTHOR]