Gap Sets for the Spectra of Cubic Graphs

Understand

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs.

• It is known that the gap intervals $(2 \sqrt{2},3)$ and $[-3,-2)$ achieved in cubic Ramanujan graphs and line graphs are maximal.
• We give constraints on spectra in [-3,3] which are maximally gapped and construct examples which achieve these bounds.
• These graphs yield new instances of maximally gapped intervals.