# On the eigenvalues of weighted directed graphs

## TL;DR

This paper explores the spectral properties of weighted directed graphs by introducing a self-adjoint operator and analyzing how graph perturbations influence eigenvalues, extending matrix analysis techniques to graph Laplacians.

## Contribution

It introduces a new self-adjoint operator for weighted directed graphs and generalizes matrix analysis methods to study eigenvalue behavior under graph perturbations.

## Key findings

- Eigenvalues are affected by graph perturbations.
- A new self-adjoint operator is proposed for spectral analysis.
- Techniques from matrix analysis are extended to directed graph Laplacians.

## Abstract

This paper deals with spectral graph theory issues related to questions of monotonicity and comparison of eigenvalues. We consider finite directed graphs with non symmetric edge weights and we introduce a special self-adjoint operator as the sum of two non self-adjoint Laplacians. We investigate how the perturbation of the graph can affect the eigenvalues. Our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized for graph Laplacians.