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AT4 family and 2-homogeneous graphs. (English) Zbl 1014.05075
Summary: Let $$\Gamma$$ denote an antipodal distance-regular graph of diameter four, with eigenvalues $$k= \theta_0> \theta_1>\cdots> \theta_4$$ and antipodal class size $$r$$. Then its Krein parameters satisfy $q^2_{11} q^3_{12} q^4_{13} q^2_{22} q^4_{22} q^3_{23} q^4_{24} q^4_{33}> 0,\quad q^2_{12}= q^4_{12}= q^4_{14}= q^3_{22}= q^4_{23}= q^4_{34}= 0$ and $q^1_{11}, q^3_{11}, q^3_{13}, q^3_{33}\in (r- 2) \mathbb{R}^+.$ It remains to consider only two more Krein bounds, namely $$q^4_{11}\geq 0$$ and $$q^4_{44}\geq 0$$. A. Jirišić and J. Koolen [Discrete Math. 244, 181-202 (2002; Zbl 1024.05086)] showed that vanishing of the Krein parameter $$q^4_{11}$$ of $$\Gamma$$ implies that $$\Gamma$$ is 1-homogeneous in the sense of K. Nomura [J. Comb. Theory, Ser. B 60, 63-71 (1994; Zbl 0793.05130)], so it is also locally strongly regular. We study vanishing of the Krein parameter $$q^4_{44}$$ of $$\Gamma$$. In this case a well-known result of P. J. Cameron et al. [J. Algebra 55, 257-280 (1978; Zbl 0444.05045)] implies that $$\Gamma$$ is locally strongly regular. We gather some evidence that vanishing of the Krein parameter $$q^4_{44}$$ implies $$\Gamma$$ is either triangle-free (in which case it is 1-homogeneous) or the Krein parameter $$q^4_{11}$$ vanishes as well. Then we prove that the vanishing of both Krein parameters $$q^4_{11}$$ and $$q^4_{44}$$ 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.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
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