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We discuss thick regular near 2d-gons with a Q-polynomial collinearity graph. For da parts per thousand yen4, we show that apart from Hamming near polygons and dual polar spaces there are no thick Q-polynomial regular near polygons. We also show that no regular near hexagons exist with parameters (s, t (2), t) equal to (3, 1, 34), (8, 4, 740), (92, 64, 1314560), (95, 19, 1027064) or (105, 147, 2763012). Such regular near hexagons are necessarily Q-polynomial. All these nonexistence results imply the nonexistence of distance-regular graphs with certain classical parameters. We also discuss some implications for the classification of dense near polygons with four points per line. |
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