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

TL;DR

This paper introduces new bounds for the normalized Laplacian Estrada and normalized energy index of graphs, using majorization techniques to improve existing inequalities based on eigenvalue localization.

Contribution

It presents novel inequalities for graph invariants derived from the eigenvalues of the normalized Laplacian, enhancing the precision of bounds through a unified approach.

Findings

New bounds for normalized Laplacian Estrada index

Sharper inequalities for normalized energy index

Numerical examples demonstrate improved bounds

Abstract

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 unified 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.