Subcubic graphs without eigenvalues in) -1, 1)

Guo and Royle recently classified the connected cubic graphs
whose adjacency spectra avoid the open interval (-1,1), and asked
whether this classification could be extended to graphs of maximum
degree at most three. In this talk, we present a complete resolution
of this question for the non-cubic subcubic case. We determine all
connected subcubic graphs that are not cubic and have no eigenvalues
in (-1,1), showing that exactly two infinite families and eight
sporadic examples occur, with every sporadic graph having at most 18
vertices. As a consequence, we prove that (-1,1) is a maximal spectral
gap set for the class of connected subcubic graphs. This strictly
extends the result of Guo and Royle, who established maximality for
connected cubic graphs.