On the A-spectra of graphs

Abstract 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 A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . In this paper, we give some results on the eigenvalues of A α ( G ) for α > 1 / 2 . In particular, we characterize the graphs with λ k ( A α ( G ) ) = α n − 1 for 2 ≤ k ≤ n . Moreover, we show that λ n ( A α ( G ) ) ≥ 2 α − 1 if G contains no isolated vertices.