Ordering trees by -index

Let G be a graph with adjacency matrix A(G) and diagonal matrix of degrees Deg(G). For every real number alpha is an element of [0, 1], let A(alpha)(G) = alpha Deg(G) + (1 - alpha)A(G). In this work, we consider the problem of ordering trees according to the spectral radius of the A alpha-matrices, the alpha-index, determining those of order n that attain from the third to sixth largest values for this parameter. In particular, these results generalize the already known ordering when the spectral radius of the adjacency or Laplacian matrices are considered, instead of the alpha-index. We also present several results concerning the alpha-index of trees when the diameter or maximum degree are fixed parameters.