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

We give a complete characterization of the cubic graphs with no eigenvalues in the open interval (−1,1). We first classify the connected cubic graphs with no eigenvalues in (−1,1) showing that there are two infinite families: one due to Guo and Mohar (2014) [7] and the other due to Kollár and Sarnak (2021) [12], and 13 “sporadic” graphs on at most 32 vertices. Then a not necessarily connected cubic graph has no eigenvalues in (−1,1) if and only if the same is true for every connected component. This classification allows us to show that (−1,1) is a maximal spectral gap set for cubic graphs, thereby answering a question of Kollár and Sarnak (2021) [12]. The techniques used include examination of the small subgraphs that can appear in such a graph and an application of the classification of generalized line graphs.