Novel Bounds for the Normalized Laplacian Estrada Index and Normalized Laplacian Energy

For a simple and connected graph, several lower and upper bounds of graph invariants expressed in terms of the eigenvalues of the normalized Laplacian Matrix have been proposed in literature. In this paper, through a unied approach based on majorization techniques, we provide some novel inequalities depending on additional information on the localization of the eigenvalues of the normalized Laplacian matrix. Some numerical examples show how sharper results can be obtained with respect to those existing in literature.