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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
Citations:
Zbl 1054.05101
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References:
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