# Dynamical systems on graphs through the signless Laplacian matrix

## TL;DR

This paper explores how the signless Laplacian matrix of graphs influences the behavior of dynamical systems, combining theoretical eigenvalue analysis with numerical experiments on real-like graphs.

## Contribution

It provides new theoretical insights into the eigenvalues of the signless Laplacian and introduces a metric linking graph topology, matrix properties, and system dynamics.

## Key findings

- Eigenvalues of the signless Laplacian relate to dynamical behavior
- Numerical results confirm theoretical predictions
- A new metric links graph rigidity to system dynamics

## Abstract

There is a deep and interesting connection between the topological properties of a graph and the behaviour of the dynamical system defined on it. We analyse various kind of graphs, with different contrasting connectivity or degree characteristics, using the signless Laplacian matrix. We expose the theoretical results about the eigenvalue of the matrix and how they are related to the dynamical system. Then, we perform numerical computations on real-like graphs and observe the resulting system. Comparing the theoretical and numerical results we found a perfect consistency. Furthermore, we define a metric which takes in account the "rigidity" of the graph and enables us to relate all together the topological properties of the graph, the signless Laplacian matrix and the dynamical system.