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Gap sets for the spectra of cubic graphs
Author(s) -
Alicia J. Kollár,
Peter Sarnak
Publication year - 2021
Publication title -
communications of the american mathematical society
Language(s) - English
Resource type - Journals
ISSN - 2692-3688
DOI - 10.1090/cams/3
Subject(s) - algorithm , artificial intelligence , computer science , materials science
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 ) (2 \sqrt {2},3) and [ − 3 , − 2 ) [-3,-2) achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in [ − 3 , 3 ] [-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 ) [-3,3) can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.

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