{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T13:22:15Z","timestamp":1775481735104,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>To each coherent configuration (scheme) ${\\cal C}$ and positive integer $m$ we associate a natural scheme $\\widehat{\\cal C}^{(m)}$  on the $m$-fold Cartesian product of the point set of ${\\cal C}$ having the same automorphism group as ${\\cal C}$. Using this construction we define and study two positive integers: the separability number $s({\\cal C})$ and the Schurity number $t({\\cal C})$ of ${\\cal C}$. It turns out that $s({\\cal C})\\le m$ iff ${\\cal C}$ is uniquely determined up to isomorphism by the intersection numbers of the scheme $\\widehat{\\cal C}^{(m)}$. Similarly, $t({\\cal C})\\le m$ iff the diagonal subscheme of $\\widehat{\\cal C}^{(m)}$ is an orbital one. In particular, if ${\\cal C}$ is the scheme of a distance-regular graph $\\Gamma$, then $s({\\cal C})=1$ iff $\\Gamma$ is uniquely determined by its parameters whereas $t({\\cal C})=1$ iff $\\Gamma$ is distance-transitive.  We show that if ${\\cal C}$ is a Johnson, Hamming or Grassmann scheme, then $s({\\cal C})\\le 2$ and $t({\\cal C})=1$. Moreover, we find the exact values of $s({\\cal C})$ and $t({\\cal C})$ for the scheme ${\\cal C}$ associated with any distance-regular graph having the same parameters as some Johnson or Hamming graph. In particular, $s({\\cal C})=t({\\cal C})=2$ if ${\\cal C}$ is the scheme of a Doob graph. In addition, we prove that $s({\\cal C})\\le 2$ and $t({\\cal C})\\le 2$ for any imprimitive 3\/2-homogeneous scheme. Finally, we show that $s({\\cal C})\\le 4$, whenever ${\\cal C}$ is a cyclotomic scheme on a prime number of points.<\/jats:p>","DOI":"10.37236\/1509","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:04:41Z","timestamp":1578690281000},"source":"Crossref","is-referenced-by-count":14,"title":["Separability Number and Schurity Number of Coherent Configurations"],"prefix":"10.37236","volume":"7","author":[{"given":"Sergei","family":"Evdokimov","sequence":"first","affiliation":[]},{"given":"Ilia","family":"Ponomarenko","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2000,5,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r31\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T00:23:21Z","timestamp":1579307001000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v7i1r31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,5,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2000,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1509","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2000,5,17]]},"article-number":"R31"}}