Abstract
It is known that a distance-regular graph with valency k at least three admits at most two Q-polynomial 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 \(\tilde H(2d + 1,2)\), or the dual polar graph on [2 A 2d-1(q)] with q ⩾ 2 a prime power.
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Ma, J., Koolen, J.H. Twice Q-polynomial distance-regular graphs of diameter 4. Sci. China Math. 58, 2683–2690 (2015). https://doi.org/10.1007/s11425-014-4958-0
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DOI: https://doi.org/10.1007/s11425-014-4958-0