Twice Q-polynomial distance-regular graphs of diameter 4 |
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Authors: | JianMin Ma Jack H Koolen |
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Institution: | 1. College of Mathematics & Information Science and Hebei Key Lab of Computational Mathematics & Applications, Hebei Normal University, Shijiazhuang, 050016, China 2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China
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Abstract: | It is known that a distance-regular graph with valency k at least three admits at most two Qpolynomial structures. We show that all distance-regular graphs with diameter four and valency at least three admitting two Q-polynomial structures are either dual bipartite or almost dual bipartite. By the work of Dickie(1995) this implies that any distance-regular graph with diameter d at least four and valency at least three admitting two Q-polynomial structures is, provided it is not a Hadamard graph, either the cube H(d, 2)with d even, the half cube 1/2H(2d + 1, 2), the folded cube?H(2d + 1, 2), or the dual polar graph on 2A2d-1(q)]with q 2 a prime power. |
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