A class of distance-regular graphs that are Q-polynomial

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We consider a class of distance-regular graphs Γ with diameter d whose intersection numbers take the form k = hxyd(t − 1) −1 b i = h(i − t)(i − x)(i − y)(i − d)(2i − t) −1(2i − t + 1) −1, (1⩽i⩽d−1) c i = hi(i − t + x)(i − t + y)(i − t + d)(2i − t)(2i − t − 1) −1, (1⩽i⩽d−1) c d = hd(d − t + x)(d − t + y)(2d − t + 1) −1 for some complex constants h, t, x, and y. We show the eigenvalues of Γ are integers if d≥4, and that d≥14 implies Γ is either • (i) the antipodal quotient of the Johnson graph J(2 d, 4 d) or J(2 d + 1, 4 d + 2) • (ii) the halved graph 1 2 H(2d + 1, 2) of the 2 d + 1-cube • (iii) the antipodal quotient of 1 2 H(4d, 2) or 1 2 H(4d + 2, 2) • (iv) a graph not listed above, but with the same intersection numbers as (i) or (iii) In particular, for d≥14, the list of known graphs of type 2 in Theorem 5.1 of Bannai and Ito (Benjamin-Cummings Lecture Note Series, Vol. 58) is complete if and only if there are no graphs satisfying (iv) above.