Abstract Recently, EEG recording techniques and source analysis have improved, making it feasible to tap into fast network dynamics. Yet, analyzing whole-cortex EEG signals in source space is not standard, partly because EEG suffers from volume conduction: Functional connectivity (FC) reflecting genuine functional relationships is impossible to disentangle from spurious FC introduced by volume conduction. Here, we investigate the relationship between white matter structural connectivity (SC) and large scale network structure encoded in EEG-FC. We start by confirming that FC (power envelope correlations) is predicted by SC beyond the impact of Euclidean distance, in line with the assumption that SC mediates genuine FC. We then use information from white matter structural connectivity (SC) in order to smooth the EEG signal in the space spanned by graphs derived from SC. Thereby, FC between nearby, structurally connected brain regions increases while FC between non-connected regions remains unchanged, resulting in an increase in genuine, SC-mediated FC. We analyze the induced changes in FC, assessing the resemblance between EEG- and volume-conduction-free fMRI-FC, and find that smoothing increases resemblance in terms of overall correlation and community structure. This result suggests that our method boosts genuine FC, an outcome that is of interest for many EEG network neuroscience questions. Author summary In this study, we combine high-density EEG recorded during resting state with white matter connectivity obtained from diffusion MRI and fiber tracking. We leverage the additional information contained in the structural connectome towards augmenting the source level EEG functional connectivity. In particular, it is known - and confirmed in this study - that the activity of brain regions that possess a direct anatomical connection is, on average, more strongly correlated than that of regions that have no such direct link. We use the structural connectome to define a graph and smooth the source reconstructed EEG signal in the space spanned by this graph. We compare the resulting “filtered” signal correlation matrices to those obtained from fMRI and find that such “graph filtering” improves the agreement between EEG and fMRI functional connectivity structure. This suggests that structural connectivity can be used to attenuate some of the limitations imposed by volume conduction.