Explicit construction of a Ramanujan (n1,n2,…,nd−1)-regular hypergraph

Abstract

Using the main properties of the skew polynomial rings ${\mathbb{F}}_{q^d}\{\tau \}$ and some related rings, we describe the explicit construction of Ramanujan hypergraphs, which are certain simplicial complexes introduced in the author's thesis [29] (see also [30]) as generalizations of Ramanujan graphs. Such hypergraphs are described in terms of Cayley graphs of various groups. We give an explicit description of our hypergraph as the Cayley graph of the groups ${\mathrm{PSL}}_d({\mathbb{F}}_r)$ and ${\mathrm{PGL}}_d({\mathbb{F}}_r)$ with respect to a certain set of generators, over a finite field ${\mathbb{F}}_r$ with $r$ elements