# Spectra of Laplacian matrices of weighted graphs: structural genericity properties

## TL;DR

This paper investigates how perturbing edge weights in weighted graphs' Laplacian matrices affects their spectra, ensuring simple eigenvalues and non-zero Fiedler vectors, which enhances understanding of complex network dynamics.

## Contribution

It demonstrates that weight perturbations can achieve spectral simplicity and non-zero Fiedler vectors without adding edges, strengthening classical genericity results.

## Key findings

- Eigenvalues can be made simple through weight perturbations.
- Fiedler vectors can be composed of only non-zero entries.
- Structural perturbations are constrained to existing edges.

## Abstract

This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its associated eigenvector, the so-called Fiedler vector. Here we prove that, given a Laplacian matrix, it is possible to perturb the weights of the existing edges in the underlying graph in order to obtain simple eigenvalues and a Fiedler vector composed of only non-zero entries. These structural genericity properties with the constraint of not adding edges in the underlying graph are stronger than the classical ones, for which arbitrary structural perturbations are allowed. These results open the opportunity to understand the impact of structural changes on the dynamics of complex systems.