Introduction

Graphs provide a powerful mathematical framework for modelling complex systems, from molecular structures to social networks. In many physical and geometric problems, nodes represent particles, and edges encode interactions, often acting like springs. This perspective aligns naturally with Geometric Deep Learning, where learning algorithms leverage graph structures to capture spatial and relational patterns.

Understanding energy functions and the forces derived from them is fundamental to modelling such systems. In physics and computational chemistry, harmonic potentials, which penalise deviations from equilibrium positions, are widely used to describe elastic networks, protein structures, and even diffusion processes. The Laplacian matrix plays a key role in these formulations, linking energy minimisation to force computations in a clean and computationally efficient way.

By formalising these interactions using matrix notation, we gain not only a compact representation but also a foundation for more advanced techniques such as Langevin dynamics, normal mode analysis, and graph-based neural networks for physical simulations.