Two-graphs and skew two-graphs in finite geometries
type: Item
instance of: scholarly article
title: Two-graphs and skew two-graphs in finite geometries
zbMATH Open document ID: 0839.05024
DOI: 10.1016/0024-3795(95)00242-J
published in: Linear Algebra and its Applications
publication date: 1996-06-18T00:00:00Z
review text: Let \(X\) be a set of cardinality \(v\geq 3\). A two-graph on \(X\) is a set, \(\Delta\), of 3-subsets of \(X\) such that every 4-subset of \(X\) contains an even number (i.e., 0, 2 or 4) of triples from \(\Delta\). A skew (oriented) two-graph on \(X\) is a set, \(\nabla\), of 3-cycles in \(\text{Sym}(X)\), such that precisely one of any 3-cycle or its inverse belongs to \(\nabla\), and for any 4-subset of \(X\), \(\nabla\) contains an even number of the 3-cycles that can be formed from these four elements. Given a 2-subset of \(X\) (resp. an ordered pair of distinct elements of \(X\)) its degree is the number of 3-subsets of \(\Delta\) (resp. 3-cycles of \(\nabla\)) which contain it. The multiset of the number of 2-subsets (resp. ordered pairs) of each degree is called the fingerprint of \(\Delta\) (resp. \(\nabla\)). In this paper, fingerprints are used as isomorphic invariants for translation planes and \(m\)-systems (including ovoids and spreads) of polar spaces in odd characteristic. The adventage of using fingerprints in this way is that they are easily and quickly computed, however, they are not generally complete, i.e., able to distinguish between any two inequivalent examples. The question of completeness of fingerprint invariants for ovoids in \(O_{2r}^+(q)\) (for \(q\) odd) is open for \(r\geq 3\), but is false for \(r= 2\) (Theorem 7.3). The author also examines caps in polar spaces whose associated two-graphs are trivial, classifying these in a small case and obtaining a general bound on the size of the cap. These results are used in obtaining a bound on the size of a family of doubly intersecting circles in classical Laguerre planes of odd order.
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51A50
Mathematics Subject Classification ID: 51E99
zbMATH DE Number: 803759
zbMATH Keywords: finite geometries
zbMATH Keywords: two-graph
zbMATH Keywords: fingerprint
zbMATH Keywords: translation planes
zbMATH Keywords: \(m\)-systems
zbMATH Keywords: spreads
zbMATH Keywords: ovoids
zbMATH Keywords: caps
zbMATH Keywords: polar spaces
zbMATH Keywords: Laguerre planes
author: G. Eric Moorhouse
reviewed by: William E. Cherowitzo
describes a project that uses: nauty
MaRDI profile type: Publication
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full work available at URL: 0024-3795(95)00242-j
OpenAlex ID: W2024941132
Wikidata QID: Q126748078
description: scientific article; zbMATH DE number 803759