  We compute the entanglement entropy in a composite system separated by a
finitely ramified boundary with the structure of a self-similar lattice graph.
We derive the entropy as a function of the decimation factor which determines
the spectral dimension, the latter being generically different from the
topological dimension. For large decimations, the graph becomes increasingly
dense, yielding a gain in the entanglement entropy which, in the asymptotically
smooth limit, approaches a constant value. Conversely, a small decimation
factor decreases the entanglement entropy due to a large number of spectral
gaps which regulate the amount of information crossing the boundary. In line
with earlier studies, we also comment on similarities with certain holographic
formulations. Finally, we calculate the higher order corrections in the
entanglement entropy which possess a log-periodic oscillatory behavior.
