An Interlacing Approach for Bounding the Sum of Laplacian Eigenvalues of Graphs

TL;DR

This paper introduces an interlacing technique to derive bounds for the sums of Laplacian eigenvalues of graphs, generalizing existing theorems and linking eigenvalues to graph parameters like connectivity.

Contribution

It presents a novel interlacing approach for bounding Laplacian eigenvalues and characterizes conditions for equality, extending classical theorems in spectral graph theory.

Findings

Derived new bounds for Laplacian eigenvalue sums

Generalized classical theorems by Grone and Merris

Connected eigenvalue bounds to graph parameters like edge-connectivity

Abstract

We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone and Merris. As a consequence we obtain inequalities involving bounds for some well-known parameters of a graph, such as edge-connectivity, and the isoperimetric number.