Eigenvalues of graph Laplacians via rank-one perturbations

TL;DR

This paper investigates how the eigenvalues of graph Laplacians are affected by rank-one perturbations, providing new proofs of classical theorems and deriving spectral properties for various graph families.

Contribution

It introduces a novel approach to analyze Laplacian spectra under rank-one perturbations, simplifying proofs and extending spectral characterizations of specific graph classes.

Findings

New proof of Kirchhoff's Matrix Tree Theorem

Spectral characterizations for complete and multipartite graphs

Explicit characteristic polynomials for threshold graphs

Abstract

We show how the spectrum of a graph Laplacian changes with respect to a certain type of rank-one perturbation. We apply our finding to give new short proofs of the spectral version of Kirchhoff's Matrix Tree Theorem and known derivations for the characteristic polynomials of the Laplacians for several well known families of graphs, including complete, complete multipartite, and threshold graphs.