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

TL;DR

This paper introduces new bounds for a graph invariant based on normalized Laplacian eigenvalues, improving the precision of existing bounds through majorization techniques and numerical validation.

Contribution

It adapts a theoretical method using majorization to localize eigenvalues, deriving sharper bounds for the sum of eigenvalue powers of normalized Laplacian matrices.

Findings

Derived tighter bounds for $s_{eta}^{*}(G)$ using majorization.

Numerical examples demonstrate improved bounds over previous results.

Enhanced understanding of eigenvalue distributions in graph spectra.

Abstract

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.