摘要
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.
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.
基金
supported by Natural Science Foundation of Hebei Province(Grant No.A2012205079)
Science Foundation of Hebei Normal University(Grant No.L2011B02)
the 100 Talents Program of the Chinese Academy of Sciences for support
