Graphs Whose Aα-Spectral Radius Does Not Exceed 2

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