On the sum of powers of the Aα-eigenvalues of graphs

Let A(G) and D(G) be the adjacency matrix and the degree diagonal matrix of a graph G, respectively. For any real number alpha is an element of[0, 1], Nikiforov recently defined the A(alpha)-matrix of G as A(alpha)(G) = alpha D(G) + (1 - alpha)A(G). The graph invariant S-alpha(p) (G) is the sum of the p-th power of the A(alpha)-eigenvalues of G for 1/2 < alpha < 1, which has a close relation to the alpha-Estrada index. In this paper, we establish some bounds on S-alpha(p) (G) and characterize the extremal graphs. In particular, we present some bounds on S-alpha(p) (G) in terms of the degree sequences, order and size of G by using majorization techniques. Moreover, we give lower and upper bounds for S-alpha(p) (G) of a bipartite graph and characterize the extremal graphs.