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 alpha is an element of [0, 1], Nikiforov [8] defined the matrix A(alpha) (G) as A(alpha )(G) = alpha D(G) + (1 - alpha) A(G). In this paper, we give some results on the eigenvalues of A(alpha)(G) for alpha > 1/2. In particular, we characterize the graphs with lambda(k) (A(alpha)(G)) = alpha n - 1 for 2 <= k <= n. Moreover, we show that lambda(n) (A(alpha)(G)) >= 2 alpha - 1 if G contains no isolated vertices. (C) 2018 Elsevier Inc. All rights reserved.