Remarks on Bounds of Normalized Laplacian Eigenvalues of Graphs

TL;DR

This paper derives bounds for the eigenvalues of the normalized Laplacian matrix of a connected graph using the number of vertices, edges, and the Randić index, enhancing understanding of spectral graph properties.

Contribution

It introduces new bounds for normalized Laplacian eigenvalues based on graph parameters and the Randić index, extending spectral graph theory insights.

Findings

Bounds for eigenvalues in terms of n and R_{-1}

Improved understanding of spectral properties of graphs

Connections between graph structure and Laplacian spectrum

Abstract

Let $G$ be a connected undirected graph with $n$, $n\ge 3$, vertices and $m$ edges. Denote by $\rho_1 \ge \rho_2 \ge \cdots > \rho_n =0$ the normalized Laplacian eigenvalues of $G$. Upper and lower bounds of $\rho_i$, $i=1,2,\ldots , n-1$, are determined in terms of $n$ and general Randi\' c index, $R_{-1}$.