  For a simple and connected graph, a new graph invariant $s_{\alpha}^{*}(G)$,
defined as the sum of powers of the eigenvalues of the normalized Laplacian
matrix, has been introduced by Bozkurt and Bozkurt in [7]. Lower and upper
bounds have been proposed by the authors. In this paper, we localize the
eigenvalues of the normalized Laplacian matrix by adapting a theoretical
method, proposed in Bianchi and Torriero ([5]), based on majorization
techniques. Through this approach we derive upper and lower bounds of
$s_{\alpha}^{*}(G)$. Some numerical examples show how sharper results can be
obtained with respect to those existing in literature.
