Projective Rectangles: The Graph of Lines
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929421966901248 |
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| author | Flórez, Rigoberto Zaslavsky, Thomas |
| author_facet | Flórez, Rigoberto Zaslavsky, Thomas |
| contents | A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique structure. The main construction of projective rectangles, stated in a previous paper, gives rectangles whose graph of lines is a known strongly regular bilinear forms graph. That fact leads to a proof that the main construction does produce projective rectangles, and also gives a new representation of bilinear forms graphs. We conclude by mentioning a few simple graph properties, such as the chromatic number, which is not known, and a partial geometry obtained from the graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11285 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective Rectangles: The Graph of Lines Flórez, Rigoberto Zaslavsky, Thomas Combinatorics 51E26 (Primary) 51A30, 51A45 (Secondary) A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique structure. The main construction of projective rectangles, stated in a previous paper, gives rectangles whose graph of lines is a known strongly regular bilinear forms graph. That fact leads to a proof that the main construction does produce projective rectangles, and also gives a new representation of bilinear forms graphs. We conclude by mentioning a few simple graph properties, such as the chromatic number, which is not known, and a partial geometry obtained from the graph. |
| title | Projective Rectangles: The Graph of Lines |
| topic | Combinatorics 51E26 (Primary) 51A30, 51A45 (Secondary) |
| url | https://arxiv.org/abs/2407.11285 |