Graph generative modelling has become an essential task due to the wide range of applications in chemistry, biology, social networks, and knowledge representation. In this work, we propose a novel framework for generating graphs by adapting the Generator Matching (arXiv:2410.20587) paradigm to graph-structured data. We leverage the graph Laplacian and its associated heat kernel to define a continous-time diffusion on each graph. The Laplacian serves as the infinitesimal generator of this diffusion, and its heat kernel provides a family of conditional perturbations of the initial graph. A neural network is trained to match this generator by minimising a Bregman divergence between the true generator and a learnable surrogate. Once trained, the surrogate generator is used to simulate a time-reversed diffusion process to sample new graph structures. Our framework unifies and generalises existing diffusion-based graph generative models, injecting domain-specific inductive bias via the Laplacian, while retaining the flexibility of neural approximators. Experimental studies demonstrate that our approach captures structural properties of real and synthetic graphs effectively.