Topological models for classical configurations
type: Item
instance of: scholarly article
title: Topological models for classical configurations
zbMATH Open document ID: 0973.05014
DOI: 10.1016/S0378-3758(99)00122-6
published in: Journal of Statistical Planning and Inference
publication date: 2001-09-13T00:00:00Z
review text: Let \(S_k\) be an orientable surface (a sphere with \(k\) ``handles'' attached), \(k\geq 0\). The Fano, Pappus and Desargues configurations are considered. The Menger graph have the points of a configuration as the vertices of the graph, and two vertices be adjacent if the points belong to a common line. The Menger graph of the Fano configuration is the complete graph \(K_7\). Any imbedding of \(K_7\) on the torus \(S_1\) models the Fano plane. As there is no imbedding of the Fano plane into \(S_0\), we say that this geometry has genus one. The Menger graph of the Pappus configuration is the complete multipartite graph \(K_{3\times 3}\). The Coxeter imbedding of \(K_{3\times 3}\) on the torus models the Pappus configuration. The Menger graph of the Desargues configuration is the triangular graph \(T(5)\) (the complement of the Petersen graph). An imbedding of \(T(5)\) into \(S_2\) is established (there is no imbedding of \(T(5)\) into \(S_1\)).
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 05E30
zbMATH DE Number: 1471124
zbMATH Keywords: configurations
zbMATH Keywords: strongly regular graphs
author: Ramon M. Figueroa-Centeno
author: Arthur T. White
reviewed by: Alexandre A. Makhnev
MaRDI profile type: Publication
cites work: Q4873725
cites work: Self-dual configurations and regular graphs
cites work: Q3757929
cites work: Q4858065
cites work: Q3344220
cites work: Efficient Imbeddings of Finite Projective Planes
full work available at URL: s0378-3758(99)00122-6
OpenAlex ID: W2068227274
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description: scientific article; zbMATH DE number 1471124