On the Aα-spectra of graphs

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α∈[0,1], Nikiforov [8] defined the matrix Aα(G) asAα(G)=αD(G)+(1−α)A(G). In this paper, we give some results on the eigenvalues of Aα(G) for α>1/2. In particular, we characterize the graphs with λk(Aα(G))=αn−1 for 2≤k≤n. Moreover, we show that λn(Aα(G))≥2α−1 if G contains no isolated vertices.