Duke Mathematical Journal

Explicit construction of a Ramanujan $(n_1,n_2,\ldots,n_{d-1})$-regular hypergraph

Alireza Sarveniazi

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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

Article information

Source
Duke Math. J. Volume 139, Number 1 (2007), 141-171.

Dates
First available in Project Euclid: 13 July 2007

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1184341240

Digital Object Identifier
doi:10.1215/S0012-7094-07-13913-9

Mathematical Reviews number (MathSciNet)
MR2322678

Subjects
Primary: 11B75: Other combinatorial number theory 11F72: Spectral theory; Selberg trace formula 11R58: Arithmetic theory of algebraic function fields [See also 14-XX] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 51E24: Buildings and the geometry of diagrams

Citation

Sarveniazi, Alireza. Explicit construction of a Ramanujan ( n 1 , n 2 , … , n d − 1 ) -regular hypergraph. Duke Math. J. 139 (2007), no. 1, 141--171. doi:10.1215/S0012-7094-07-13913-9. http://projecteuclid.org/euclid.dmj/1184341240.


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