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On bipartite Q-polynomial distance-regular graphs with c [sub] 2 [equal] 1Štefko Miklavič, 2007, original scientific article

**Abstract:** Let ▫$\Gamma$▫ denote a bipartite ▫$Q$▫-polynomial distance-regular graph with diameter ▫$d \ge 3$▫, valency ▫$k \ge 3$▫ and intersection number ▫$c_2=1$▫. We show that ▫$\Gamma$▫ has a certain equitable partition of its vertex set which involves ▫$4d-4$▫ cells. We use this partition to show that the intersection numbers of ▫$\Gamma$▫ satisfy the following divisibility conditions: (I) ▫$c_{i+1}-1$▫ divides ▫$c_i(c_i-1)$▫ for ▫$2 \le i \le d-1$▫, and (II) ▫$b_{i-1}-1$▫ divides ▫$b_i(b_i-1)$▫ for ▫$1 \le i \le d-1$▫. Using these divisibility conditions we show that ▫$\Gamma$▫ does not exist if ▫$d=4$▫.

**Found in:** ključnih besedah

**Summary of found:** ...We show that ▫$\Gamma$▫ has a certain equitable partition of its vertex set which involves...

**Keywords:** mathematics, grah theory, distance-regular graphs, ▫$Q$▫-polynomial property, equitable partitions

**Published:** 15.10.2013; **Views:** 1828; **Downloads:** 15

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