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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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