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 alpha is an element of [0, 1], Nikiforov defined the A(alpha)-matrix of G as A(alpha)(G) = alpha D(G) + (1 - alpha)A(G). The eigenvalues of the matrix A(alpha)(G) form the A(alpha)-spectrum of G. The A(alpha)-spectral radius of G is the largest eigenvalue of A(alpha)(G) denoted by p alpha(G). In this paper, we propose the A alpha--matrix of G as A(alpha)-(G) = alpha D(G) + (alpha - 1)A(G), 0 < alpha < 1. Let the A(alpha)--spectral radius of G be denoted by il alpha-(G), and let Sf alpha(G) and S alpha- f (G) be the sum of the fth powers of the A(alpha)and A(alpha)- eigenvalues of G, respectively. We determine the A(alpha)--spectra of some graphs and obtain some bounds of the A(alpha)--spectral radius. Moreover, we establish a relationship between the A(alpha)-spectral radius and A(alpha)--spectral radius. Indeed, for alpha is an element of (21,1), we show that il alpha- < p alpha, 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 il alpha-(G), and we prove that the A(alpha)--spectrum and A(alpha)-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 alpha is an element of [21, 1), if 0 < f< 1 or 2 < f < 3, then Sf alpha(G) >= S alpha- f (G), and if 1 < f < 2, then Sf alpha(G) < S alpha- f (G), where the equality holds if and only if G is a bipartite graph such that f g {1, 2, 3}.