AT4 family and 2-homogeneous graphs Original Research Article

Let Γ denote an antipodal distance-regular graph of diameter four, with eigenvalues k=θ0>θ1>⋯>θ4 and antipodal class size r. Then its Krein parameters satisfyq112 q123 q134 q222 q224 q233 q244 q334>0, q122=q124=q144=q223=q234=q344=0andq111,q113,q133,q333∈(r−2)R+.It remains to consider only two more Krein bounds, namely q114⩾0 and q444⩾0. Jurišić and Koolen showed that vanishing of the Krein parameter q114 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 q444 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 q444 implies Γ is either triangle-free (in which case it is 1-homogeneous) or the Krein parameter q114 vanishes as well. Then we prove that the vanishing of both Krein parameters q114 and q444 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.