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

AbstractWe consider a class of distance-regular graphs Γ with diameter d whose intersection numbers take the form k = hxyd(t − 1)−1bi = h(i − t)(i − x)(i − y)(i − d)(2i − t)−1(2i − t + 1)−1, (1⩽i⩽d−1)ci = hi(i − t + x)(i − t + y)(i − t + d)(2i − t)(2i − t − 1)−1, (1⩽i⩽d−1)cd = 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(2d, 4d) or J(2d + 1, 4d + 2)•(ii) the halved graph 12H(2d + 1, 2) of the 2d + 1-cube•(iii) the antipodal quotient of 12H(4d, 2) or 12H(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