We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals
(
2
2
,
3
)
(2 \sqrt {2},3)
and
[
−
3
,
−
2
)
[-3,-2)
achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in
[
−
3
,
3
]
[-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
)
[-3,3)
can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.
A Connection Between a Question of Bermond and Bollobás and Ramanujan Graphs
