Cubic graphs with no eigenvalues in the interval (-1,1)

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 Kollár 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.