Ordering graphs by their largest (least) Aα-eigenvalues

Let G be a simple undirected graph. For 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), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G respectively. In this paper, we obtain a sharp upper bound on the largest eigenvalue rho(alpha)(G) of A(alpha)(G) for alpha is an element of [1 /2, 1). Employing this upper bound, we prove that 'For connected G(1) and G(2) with n vertices and m edges, if the maximum degree Delta(G(1)) >= 2 alpha(1 - alpha)(2m - n + 1 ) 2 alpha and Delta(G1) > Delta(G(2)), then rho(alpha) (G(1)) > rho(alpha)(G(2))'. Let lambda(alpha)(G) denote the least eigenvalue of A(alpha)(G). For alpha is an element of (1 /2, 1), we prove that 'For two connected G(1) and G(2), if the minimum degree delta(G(1)) <= 1/1-alpha - 2 and delta(G(1)) < delta(G(2)), then lambda(alpha)(G(1)) < X lambda(alpha)(G(2))'.