Ramanujan graphs of very large girth based on octonions
Combinatorics
2012-02-06 v5
Abstract
We present a generalization of the construction of graphs by Lubotzky, Phillips and Sarnak in their celebrated article "Ramanujan graphs". The new approach consists in using octonion algebras rather than quaternions. A key tool is the existing result of the unique factorization of integral octonions. The families obtained by this mean present not only the same spectral property that make them good expanders, but also show a larger girth, yielding a new record for regular graphs.
Cite
@article{arxiv.1011.2642,
title = {Ramanujan graphs of very large girth based on octonions},
author = {Xavier Dahan and Jean-Pierre Tillich},
journal= {arXiv preprint arXiv:1011.2642},
year = {2012}
}
Comments
Paper withdrawn due to a major flaw in Lemma 6. Some computational checks show that the two key results (very large girth and Ramanujan property) do not hold in general