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Separability number and Schurity number of coherent configurations. (English) Zbl 0939.05087
Electron. J. Comb. 7, No. 1, Research paper R31, 33 p. (2000); printed version J. Comb. 7, No. 1 (2000).
Summary: To each coherent configuration (scheme) \({\mathcal C}\) and positive integer \(m\) we associate a natural scheme \(\widehat{\mathcal C}^{(m)}\) on the \(m\)-fold Cartesian product of the point set of \({\mathcal C}\) having the same automorphism group as \({\mathcal C}\). Using this construction we define and study two positive integers: the separability number \(s({\mathcal C})\) and the Schurity number \(t({\mathcal C})\) of \({\mathcal C}\). It turns out that \(s({\mathcal C})\leq m\) iff \({\mathcal C}\) is uniquely determined up to isomorphism by the intersection numbers of the scheme \(\widehat{\mathcal C}^{(m)}\). Similarly, \(t({\mathcal C})\leq m\) iff the diagonal subscheme of \(\widehat{\mathcal C}^{(m)}\) is an orbital one. In particular, if \({\mathcal C}\) is the scheme of a distance-regular graph \(\Gamma\), then \(s({\mathcal C})=1\) iff \(\Gamma\) is uniquely determined by its parameters whereas \(t({\mathcal C})=1\) iff \(\Gamma\) is distance-transitive. We show that if \({\mathcal C}\) is a Johnson, Hamming or Grassmann scheme, then \(s({\mathcal C})\leq 2\) and \(t({\mathcal C})=1\). Moreover, we find the exact values of \(s({\mathcal C})\) and \(t({\mathcal C})\) for the scheme \({\mathcal C}\) associated with any distance-regular graph having the same parameters as some Johnson or Hamming graph. In particular, \(s({\mathcal C})=t({\mathcal C})=2\) if \({\mathcal C}\) is the scheme of a Doob graph. In addition, we prove that \(s({\mathcal C})\leq 2\) and \(t({\mathcal C})\leq 2\) for any imprimitive 3/2-homogeneous scheme. Finally, we show that \(s({\mathcal C})\leq 4\), whenever \({\mathcal C}\) is a cyclotomic scheme on a prime number of points.

MSC:
05E30 Association schemes, strongly regular graphs
05C12 Distance in graphs
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