There is no McLaughlin geometry

Patric R.J. Östergård; Leonard H. Soicher

doi:10.1016/j.jcta.2017.10.004

There is no McLaughlin geometry

Patric R.J. Östergård, Leonard H. Soicher

Information Theory

Queen Mary University of London

Research output: Contribution to journal › Article › Scientific › peer-review

2 Citations (Scopus)

Abstract

We determine that there is no partial geometry G with parameters (s,t,α)=(4,27,2). The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in G is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.

Original language	English
Pages (from-to)	27-41
Number of pages	15
Journal	Journal of Combinatorial Theory. Series A
Volume	155
DOIs	https://doi.org/10.1016/j.jcta.2017.10.004
Publication status	Published - 1 Apr 2018
MoE publication type	A1 Journal article-refereed

Keywords

Backtrack search
Krein bound
McLaughlin geometry
McLaughlin graph
Partial geometry
Pseudogeometric graph

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10.1016/j.jcta.2017.10.004

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title = "There is no McLaughlin geometry",

abstract = "We determine that there is no partial geometry G with parameters (s,t,α)=(4,27,2). The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in G is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.",

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AU - Östergård, Patric R.J.

AU - Soicher, Leonard H.

PY - 2018/4/1

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N2 - We determine that there is no partial geometry G with parameters (s,t,α)=(4,27,2). The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in G is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.

AB - We determine that there is no partial geometry G with parameters (s,t,α)=(4,27,2). The existence of such a geometry has been a challenging open problem of interest to researchers for almost 40 years. The particular interest in G is due to the fact that it would have the exceptional McLaughlin graph as its point graph. Our proof makes extensive use of symmetry and high-performance distributed computing, and details of our techniques and checks are provided. One outcome of our work is to show that a pseudogeometric strongly regular graph achieving equality in the Krein bound need not be the point graph of any partial geometry.

KW - Backtrack search

KW - Krein bound

KW - McLaughlin geometry

KW - McLaughlin graph

KW - Partial geometry

KW - Pseudogeometric graph

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DO - 10.1016/j.jcta.2017.10.004

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SN - 0097-3165

VL - 155

SP - 27

EP - 41

JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

ER -

There is no McLaughlin geometry

Abstract

Keywords

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