The distance-regular graphs of valency four
type: Item
instance of: scholarly article
title: The distance-regular graphs of valency four
zbMATH Open document ID: 0929.05093
DOI: 10.1023/A:1018693118268
published in: Journal of Algebraic Combinatorics
publication date: 2000-01-25T00:00:00Z
review text: We report on a computer search that proves that each distance-regular graph of valency four has known parameters. Here we describe first the known examples, next how putative arrays were disposed of, and finally how the search could be limited to a manageable number of arrays. The distance-regular graphs of valency 3 have been determined by \textit{N. L. Biggs}, \textit{A. G. Boshier}, and \textit{J. Shawe-Taylor} [J. Lond. Math. Soc., II. Ser. 33, 385-394 (1986; Zbl 0576.05039)]. \textit{E. Bannai} and \textit{T. Itô} worked on the general project of bounding the diameter of a distance-regular graph as a function of its valency \(k\). They succeeded in the bipartite case [J. Algebra 107, 43-52 (1987; Zbl 0677.05060)] and in case \(k= 4\) [see Eur. J. Comb. 10, No. 2, 137-148 (1989; Zbl 0677.05061)]. This means that finding the feasible arrays for distance-regular graphs of valency 4 was reduced to a finite amount of work, but the diameter bounds obtained were not small enough to straightforwardly settle this case. In this note we obtain some additional conditions, and thus reduce the parameter space to be searched, and describe a way to test a parameter set using (small) integer arithmetic, thus avoiding accuracy problems.
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1319550
zbMATH Keywords: Terwilliger bound
zbMATH Keywords: eigenvalues
zbMATH Keywords: intersection array
zbMATH Keywords: distance-regular graph
author: Andries E. Brouwer
author: Jack H. Koolen
describes a project that uses: nauty
MaRDI profile type: Publication
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full work available at URL: a:1018693118268
OpenAlex ID: W2156644942
description: scientific article; zbMATH DE number 1319550