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Let G be a connected graph of order n with adjacency matrix A (G) and degree diagonal matrix D (G). The A α -matrix of G is defined as A α (G) = α D (G) + (1 − α) A (G) for any α ∈ [ 0 , 1). All the eigenvalues of A α (G) are called A α -eigenvalues of G , and the A α -spectrum of G consists of all the A α -eigenvalues along with their multiplicities. Denote the set of connected graphs (resp., of order n) with distinct A α -eigenvalues by G A α (resp., G n A α ). We also use G A α ⁎ (resp., G n A α ⁎ ) to denote the set of connected graphs in G A α (resp., G n A α ) whose A α -eigenvalues are main. Two graphs G and H are called A α -cospectral if they have the same A α -spectrum. This paper proposes a new approach to construct infinite families of G A α and G A α ⁎ . More specifically, for a graph G in G n A α or G n A α ⁎ , the infinite families of G A α or G A α ⁎ are constructed from G. At the same time, the A α -spectra of these graphs are completely determined by the A α -spectrum of G. Finally, using this technique, we also construct some infinite families of non-isomorphic A α -cospectral graphs in G A α and G A α ⁎ . Applying the results obtained to two cases of adjacency matrix and signless Laplacian matrix, we can derive the main results in [9] and [10]. [ABSTRACT FROM AUTHOR]