Finite analogues of Euclidean space
type: Item
instance of: scholarly article
title: Finite analogues of Euclidean space
zbMATH Open document ID: 0874.05030
DOI: 10.1016/0377-0427(95)00261-8
author: Yanyan Li
published in: Journal of Computational and Applied Mathematics
publication date: 1997-11-11T00:00:00Z
review text: Let \(\mathbb{F}_q\) be the finite field with \(q=p^r\) elements for \(p\) an odd prime. Define a ``distance'' on \(\mathbb{F}^n_q\) by \(d(x,y)={^t(x-y)}(x-y)\) for column vectors \(x,y\in\mathbb{F}^n_q\) and join two vertices \(x,y\in\mathbb{F}^n_q\) by an edge iff \(d(x,y)= a\), where \(a\in\mathbb{F}_q\) is fixed. This defines a graph \(E_q(n,a)\) associated with \(\mathbb{F}^n_q\). For \((q,n,a)\neq(q,2,0)\) with \(-1\) not a square in \(\mathbb{F}_q\), the graph \(E_q(n,a)\) is a connected regular graph of known degree (see Theorem 1). The eigenvalues of the adjacency operators can be expressed in terms of Gauß sums and generalized Kloosterman sums. If \(\lambda\) is an eigenvalue of the adjacency operator with \(\lambda\) different from the degree of the graph, then Weil's estimate for Kloosterman sums yields \(|\lambda|\leq 2q^{(n-1)/2}\). From this it is not hard to prove that for \(n=3\) many of the above graphs are Ramanujan and many are not Ramanujan. It is further shown that for fixed \(q\) and \(n\) the set of graphs \(E_q(n,a)\) splits into very few isomorphism classes. Hence, these finite Euclidean graphs differ from the graphs constructed for the non-Euclidean finite upper half plane. -- The paper is illustrated by some beautiful plots in Mathematica.
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 20H30
Mathematics Subject Classification ID: 05C35
zbMATH DE Number: 908394
zbMATH Keywords: Ramanujan graphs
zbMATH Keywords: adjacency operators
zbMATH Keywords: Gauß sums
zbMATH Keywords: generalized Kloosterman sums
zbMATH Keywords: finite Euclidean graphs
reviewed by: Jürgen Elstrodt
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00261-8
OpenAlex ID: W2030077266
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description: scientific article; zbMATH DE number 908394