Abstract: Flow-matching models start from an isotropic Gaussian source, the standard choice when the correlation structure of the data is unknown in advance. For multi-channel brain recordings, however, part of this structure is known in advance. Electrodes sit at fixed positions on the head, and volume conduction through the skull and scalp makes nearby electrodes co-vary in a way that is shared across subjects. Existing EEG generative models nonetheless leave the network to learn this from scratch. We put this structure into the source instead. From the sensor coordinates alone, we build a k-nearest-neighbor graph and take a graph-Matérn function of its Laplacian as the source covariance, so the flow starts from spatially coherent patterns rather than channel-independent noise. The change adds no learned parameters, works with any coupling and any drift network, and uses the same three hyperparameters on every dataset. Across eight EEG datasets and four flow-matching methods, the graph-Matérn source lowers the spectral discrepancy between generated and real signals in the five clinical bands (PSD-KL) on most datasets. PSD-KL falls by 12% to 17% in geometric mean over datasets depending on the method and by up to 40% on PhysioNet-MI, the densest montage. We show that the improvement stems from the spatial eigenvectors of the local graph of sensor positions, since randomizing the eigenvectors while preserving the eigenvalue spectrum eliminates the gain. Furthermore, a prior fitted directly to the empirical data covariance performs worse than isotropic noise. The same construction applies unchanged to MEG, intracranial EEG with patient-specific grids, and a traffic-sensor network, lowering PSD-KL for every method on each. https://jd730.github.io/projects/GraphPrior