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Let A(G) and D(G) be the adjacency matrix and the degree matrix of a graph G, respectively. For any real α ∈ [0, 1], we consider Aα (G) = αD(G) + (1 − α)A(G) as a graph matrix, whose largest eigenvalue is called the Aα -spectral radius of G. We first show that the smallest limit point for the Aα -spectral radius of graphs is 2, and then we characterize the connected graphs whose Aα -spectral radius is at most 2. Finally, we show that all such graphs, with four exceptions, are determined by their Aα -spectra.