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arXiv · 2005.05379

Gap Sets for the Spectra of Cubic Graphs

Alicia J. Kollár·Peter Sarnak
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Abstract

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. We also show that every point in $[-3,3)$ can be gapped by cubic graphs, even by planar ones. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.

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Alicia J. Kollár, Peter Sarnak. 2020-05-11. Gap Sets for the Spectra of Cubic Graphs. https://arxiv.org/abs/2005.05379

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