AT4 family and 2-homogeneous graphs

Let Gamma denote an antipodal distance-regular graph of diameter four, with eigenvalues k = theta0 > theta1 >... > theta4 and antipodal class size r. Then its Krein parameters satisfy q(11)(2) q(12)(3) q(13)(4) q(22)(2) q(22)(4) q(23)(3) q(24)(4) q(33)(4) > 0, q(12)(2) = q(12)(4) = q(14)(4) = q(22)(3) = q(23)(4) = q(34)(4) = 0 and q(11)(1), q(11)(3), q(13)(3), q(33)(3) is an element of(r - 2) R+. It remains to consider only two more Krem bounds, namely q411 greater than or equal to 0 and q444 greater than or equal to 0. Jurisic and Koolen showed that vanishing of the Krein parameter q411 of Gamma implies that Gamma is 1-homogeneous in the sense of Nomura, so it is also locally strongly regular. We study vanishing of the Krein parameter q(44)(4) of Gamma. In this case a well-known result of Cameron et al. implies that Gamma is locally strongly regular. We gather some evidence that vanishing of the Krein parameter q(44)(4) implies Gamma is either triangle-free (in which case it is 1-homogeneous) or the Krein parameter q(11)(4) vanishes as well. Then we prove that the vanishing of both Krein parameters q(11)(4) and q(44)(4) of Gamma implies that every second subconstituent graph is again an antipodal distance-regular graph of diameter four. Finally, if Gamma is also a double-cover, i.e., r = 2, i.e., Q-polynomial, then it is 2-homogeneous in the sense of Nomura. (C) 2002 Elsevier Science B.V. All rights reserved.