On the Aα--spectra of graphs and the relation between Aα- and Aα--spectra.

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 number α ∈ [0,1], Nikiforov defined the A_α-matrix of G as A_α (G) = αD(G) + (1 - α)A(G). The eigenvalues of the matrix A_α(G) form the A_α-spectrum of G. The A_α-spectral radius of G is the largest eigenvalue of A_α(G) denoted by ρ_α(G). In this paper, we propose the A_α--matrix of G as A_α-(G) = αD(G) + (α - 1)A(G), 0 ≤ α ≤ 1. Let the A_α--spectral radius of G be denoted by λ_α-(G), and let S_β^α(G) and S_β^α- (G) be the sum of the β^th powers of the A_α and A^α- eigenvalues of G, respectively. We determine the A^α--spectra of some graphs and obtain some bounds of the A^α--spectral radius. Moreover, we establish a relationship between the A^α-spectral radius and A^α--spectral radius. Indeed, for α ∈ (1/2, 1), we show that λ_α- ≤ ρα, and we prove that if G is connected, then the equality holds if and only if G is bipartite. Employing this relation, we obtain some upper bounds of λ_α- (G), and we prove that the A_α--spectrum and Aα-spectrum are equal if and only if G is a bipartite connected graph. Furthermore, we generalize the relation established by S. Akbari et al. in (2010) as follows: for α ∈ [1/2, 1), if 0 < β ≤ 1 or 2 ≤ β ≤ 3, then S_β^α(G) ≥ S_β^α- (G), and if 1 ≤ β ≤ 2, then S_β^α(G) ≤ S_β^α-(G), where the equality holds if and only if G is a bipartite graph such that β ∉ {1,2,3}. [ABSTRACT FROM AUTHOR]