New Bounds for the Sum of Powers of Normalized Laplacian Eigenvalues of\n Graphs

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