Graph reduction techniques and the multiplicity of the Laplacian eigenvalues

TL;DR

This paper extends known results on how graph modifications affect the multiplicity of Laplacian eigenvalues, providing new theoretical insights and proofs related to generalized Laplacian matrices and eigenvalue multiplicities.

Contribution

It generalizes existing theorems on Laplacian eigenvalue multiplicities and offers two proofs for a conjecture linking eigenvalue multiplicities to pendant paths.

Findings

Extended Grone et al.'s theorem to generalized Laplacian eigenvalues

Proved that certain graph modifications do not change eigenvalue multiplicities

Provided two proofs for a conjecture relating eigenvalue multiplicities and pendant paths

Abstract

Let $M=[m_{ij}]$ be an $n\times m$ real matrix, $\rho$ be a nonzero real number, and $A$ be a symmetric real matrix. We denote by $D(M)$ the $n\times n$ diagonal matrix $diag(\sum_{j=1}^{m}m_{1j},\ldots,\sum_{j=1}^{m}m_{nj})$ and denote by $L_{A}^{\rho}$ the generalized Laplacian matrix $D(A)-\rho A$. A well-known result of Grone et al. states that by connecting one of the end-vertices of $P_{3}$ to an arbitrary vertex of a graph, does not change the multiplicity of Laplacian eigenvalue $1$. We extend this theorem and some other results for a given generalized Laplacian eigenvalue $\mu$. Furthermore, we give two proofs for a conjecture by Saito and Woei on the relation between the multiplicity of some Laplacian eigenvalues and pendant paths.