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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.more » « less
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We show that the commutator equation over $$SL_2\Z$$ satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for $$SL_2(\Z[\frac{1}{p}])$$. The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces.more » « less
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