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On bipartite $$Q$$-polynomial distance-regular graphs with diameter 9, 10, or 11. (English) Zbl 1391.05174
Summary: Let $$\Gamma$$ denote a bipartite distance-regular graph with diameter $$D$$. In [Graphs Comb. 20, No. 1, 47–57 (2004; Zbl 1054.05101)] J. S. Caughman IV showed that if $$D \geq 12$$, then $$\Gamma$$ is $$Q$$-polynomial if and only if one of the following (i)-(iv) holds: (i) $$\Gamma$$ is the ordinary $$2D$$-cycle, (ii) $$\Gamma$$ is the Hamming cube $$H(D,2)$$, (iii) $$\Gamma$$ is the antipodal quotient of the Hamming cube $$H(2D,2)$$, (iv) the intersection numbers of $$\Gamma$$ satisfy $$c_i = (q^i - 1)/(q-1)$$, $$b_i = (q^D-q^i)/(q-1)$$ $$(0 \leq i \leq D)$$, where $$q$$ is an integer at least $$2$$. In this paper we show that the above result is true also for bipartite distance-regular graphs with $$D \in \{9,10,11\}$$.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05E30 Association schemes, strongly regular graphs
Zbl 1054.05101
Full Text:
##### References:
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