AT4 family and 2-homogeneous graphs

Abstract

Let Γ denote an antipodal distance-regular graph of diameter four, with eigenvalues k= θ 0> θ 1>⋯> θ 4 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∈(r−2) R +. It remains to consider only two more Krein bounds, namely q 11 4⩾0 and q 44 4⩾0. Jurišić and Koolen showed that vanishing of the Krein parameter q 11 4 of Γ implies that Γ 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 Γ. In this case a well-known result of Cameron et al. implies that Γ is locally strongly regular. We gather some evidence that vanishing of the Krein parameter q 44 4 implies Γ 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 Γ implies that every second subconstituent graph is again an antipodal distance-regular graph of diameter four. Finally, if Γ is also a double-cover, i.e., r=2, i.e., Q-polynomial, then it is 2-homogeneous in the sense of Nomura.