{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:22Z","timestamp":1753893802711,"version":"3.41.2"},"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>We propose a new elementary definition of the Higman-Sims graph in which the 100 vertices are parametrised with ${\\Bbb Z}_4\\times{\\Bbb Z}_5\\times{\\Bbb Z}_5$ and adjacencies are described by linear and quadratic equations.  This definition extends Robertson's pentagon-pentagram definition of the Hoffman-Singleton graph and is obtained by studying maximum cocliques of the Hoffman-Singleton graph in Robertson's parametrisation.  The new description is used to count the 704 Hoffman-Singleton subgraphs in the Higman-Sims graph, and to describe the two orbits of the simple group HS on them, including a description of the doubly transitive action of HS within the Higman-Sims graph.  Numerous geometric connections are pointed out. As a by-product we also have a new construction of the Steiner system $S(3,6,22)$.<\/jats:p>","DOI":"10.37236\/1830","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T21:31:27Z","timestamp":1578691887000},"source":"Crossref","is-referenced-by-count":2,"title":["On the Graphs of Hoffman-Singleton and Higman-Sims"],"prefix":"10.37236","volume":"11","author":[{"given":"Paul R.","family":"Hafner","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,11,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r77\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r77\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:53:07Z","timestamp":1579305187000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r77"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,11,3]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1830","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2004,11,3]]},"article-number":"R77"}}