AT4 Family And2-homogeneous Graphs

Let @C denote an antipodal distance-regular graph of diameter four, with eigenvalues [email protected]"0>@q"1>...>@q"4 and antipodal class size r. Then its Krein parameters satisfyq"1"1^2q"1"2^3q"1"3^4q"2"2^2q"2"2^4q"2"3^3q"2"4^4q"3"3^4>0,q"1"2^2=q"1"2^4=q"1"4^4=q"2"2^3=q"2"3^4=q"3"4^4=0andq"1"1^1,q"1"1^3,q"1"3^3,q"3"3^[email protected]?(r-2)R^+.It remains to consider only two more Krein bounds, namely q"1"1^4>=0 and q"4"4^4>=0. Jurisic and Koolen showed that vanishing of the Krein parameter q"1"1^4 of @C implies that @C is 1-homogeneous in the sense of Nomura, so it is also locally strongly regular. We study vanishing of the Krein parameter q"4"4^4 of @C. In this case a well-known result of Cameron et al. implies that @C is locally strongly regular. We gather some evidence that vanishing of the Krein parameter q"4"4^4 implies @C is either triangle-free (in which case it is 1-homogeneous) or the Krein parameter q"1"1^4 vanishes as well. Then we prove that the vanishing of both Krein parameters q"1"1^4 and q"4"4^4 of @C implies that every second subconstituent graph is again an antipodal distance-regular graph of diameter four. Finally, if @C is also a double-cover, i.e., r=2, i.e., Q-polynomial, then it is 2-homogeneous in the sense of Nomura.