For a simple and connected graph, a new graph invariant s(G), defined as the sum of alpha-powers of the eigenvalues of the normalized Laplacian matrix, has been introduced by Bozkurt and Bozkurt (2012). Lower and upper bounds for this index 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 (2000), based on majorization techniques. Through this approach we derive upper and lower bounds of s(G). Some numerical examples show how sharper results can be obtained with respect to those existing in literature.
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