×

zbMATH — the first resource for mathematics

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:
05E30 Association schemes, strongly regular graphs
PDF BibTeX XML Cite
Full Text: DOI