  We study gaps in the spectra of the adjacency matrices of large finite cubic
graphs. It is known that the gap intervals $(2 \sqrt{2},3)$ and $[-3,-2)$
achieved in cubic Ramanujan graphs and line graphs are maximal. We give
constraints on spectra in [-3,3] which are maximally gapped and construct
examples which achieve these bounds. These graphs yield new instances of
maximally gapped intervals. We also show that every point in $[-3,3)$ can be
gapped by cubic graphs, even by planar ones. Our results show that the study of
spectra of cubic, and even planar cubic, graphs is subtle and very rich.
