Gap Sets for the Spectra of Cubic Graphs

2020/05/11 by Kollár, Alicia J., Sarnak, Peter · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2005.05379

Abstract

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals (2 √(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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• Finite projective planes meet spectral gaps
• Cubic graphs with no eigenvalues in the interval (-2,0)

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