Abstract
Suppose \(\mathcal {S}\) is a finite near hexagon of order (s, t) having v points. For every \(i \in \{ 0,1,2,3\}\), let \(A_i\) denote the adjacency matrix of the graph defined on the points by the distance i relation. We perform a study of the real algebra generated by the \(A_i\)’s, and take a closer look to the structure of these algebras for all known examples of \(\mathcal {S}\). Among other things, we show that a certain number \(d_\mathcal {S}\) (which is a function of s, t and v) must be integral. This allows us to exclude certain near hexagons whose (non)existence was already open for about 15 years. In the special case \(s=2\), we also show that the embedding rank of the near hexagon is at least the number \(d_{\mathcal {S}}\), and that the near hexagon has non-full projective dimensions with vector dimension equal to \(d_{\mathcal {S}}\).
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De Bruyn, B. On the adjacency algebras of near hexagons with an order. Graphs and Combinatorics 33, 1219–1230 (2017). https://doi.org/10.1007/s00373-017-1839-7
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DOI: https://doi.org/10.1007/s00373-017-1839-7