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<p>We provide local recognition results for all infinite families of affine rank 3 graphs which are not locally a disjoint union of cliques and which are not 1-dimensional (that is, for the graphs corresponding to Liebeck’s classes (A3) up to (A10) of affine rank 3 groups). We show that the local graphs alone do not characterise the global graphs by providing counterexamples. Our principal result is that the global graph is determined by the local if one adds the condition that the number of vertices adjacent to two given vertices at distance 2 from each other is a constant. Alternative conditions for specific classes are also given.</p>

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Local recognition of affine rank 3 graphs

  • 期刊论文

  • 发表日期：

    2025年10月14日

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  • Đorđe Mitrović,
  • Jeroen Schillewaert,
  • Hendrik Van Maldeghem

摘要

We provide local recognition results for all infinite families of affine rank 3 graphs which are not locally a disjoint union of cliques and which are not 1-dimensional (that is, for the graphs corresponding to Liebeck’s classes (A3) up to (A10) of affine rank 3 groups). We show that the local graphs alone do not characterise the global graphs by providing counterexamples. Our principal result is that the global graph is determined by the local if one adds the condition that the number of vertices adjacent to two given vertices at distance 2 from each other is a constant. Alternative conditions for specific classes are also given.