Ordering trees by α-index

Let G be a graph with adjacency matrix A ( G ) and diagonal matrix of degrees Deg ( G ). For every real number α ∈ [ 0 , 1 ] , let A α ( G ) = α Deg ( G ) + ( 1 - α ) A ( G ) . In this work, we consider the problem of ordering trees according to the spectral radius of the A α -matrices, the α -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 α -index. We also present several results concerning the α -index of trees when the diameter or maximum degree are fixed parameters.