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We give a complete characterisation of the cubic graphs with no eigenvalues in the open interval (-1, 1). There are two infinite families, one due to Guo and Mohar Linear Algebra Appl. 449: 68--75 the other due to Koll\'ar and Sarnak Communications of the AMS. 1, 1--38, and 14 "sporadic" graphs on at most 32 vertices. This allows us to show that (-1, 1) is a maximal spectral gap set for cubic graphs. Our techniques including examination of various substructure and an application of the classification of generalized line graphs.
Guo et al. (Wed,) studied this question.