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 alpha is an element of [0; 1], we consider A(alpha)(G) = D-alpha(G) + (1 - alpha)A(G) as a graph matrix, whose largest eigenvalue is called the A(alpha)-spectral radius of G. We first show that the smallest limit point for the A(alpha)-spectral radius of graphs is 2, and then we characterize the connected graphs whose A(alpha)-spectral radius is at most 2. Finally, we show that all such graphs, with four exceptions, are determined by their A(alpha)-spectra.