Ramanujan graphs of very large girth based on octonions

Details Ramanujan graphs of very large girth based on octonions Ramanujan graphs of very large girth based on octonions

2010-11-11

arxiv.org/abs/1011.2642v5

Mathematics

Combinatorics

Paper withdrawn due to a major flaw in Lemma 6. Some computational checks show that the two key results (very large girth and

Scientific paper

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.

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