Minimum vertex covers and the spectrum of the normalized Laplacian on trees

TL;DR

This paper explores the relationship between the spectrum of the normalized Laplacian on trees and minimum vertex covers, revealing how eigenvalues near 1 relate to vertex cover properties.

Contribution

It establishes a novel connection between eigenvalues of the normalized Laplacian and minimum vertex covers on trees, including eigenvalue multiplicities and eigenvector entries.

Findings

Eigenvalue 1 multiplicity relates to minimum vertex cover size

Zero entries of eigenvectors correspond to vertices in minimum vertex covers

Eigenvalues near 1 can be estimated from minimum vertex covers

Abstract

We show that, in the graph spectrum of the normalized graph Laplacian on trees, the eigenvalue 1 and eigenvalues near 1 are strongly related to minimum vertex covers. In particular, for the eigenvalue 1, its multiplicity is related to the size of a minimum vertex cover, and zero entries of its eigenvectors correspond to vertices in minimum vertex covers; while for eigenvalues near 1, their distance to 1 can be estimated from minimum vertex covers; and for the largest eigenvalue smaller than 1, the sign graphs of its eigenvectors take vertices in a minimum vertex cover as representatives.