Geodesic
On the graphs of Hoffman-Singleton and Higman-Sims
The electronic journal of combinatorics, Tome 11 (2004) no. 1
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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 $S(3,6,22)$.
DOI : 10.37236/1830
Classification : 05C62, 05C25, 51E10, 51E26
Mots-clés : Higman-Sims graph, Hoffman-Singleton graph, Steiner system 
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     author = {Paul R. Hafner},
     title = {On the graphs of {Hoffman-Singleton} and {Higman-Sims}},
     journal = {The electronic journal of combinatorics},
     year = {2004},
     volume = {11},
     number = {1},
     doi = {10.37236/1830},
     zbl = {1060.05073},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1830/}
}
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Paul R. Hafner. On the graphs of Hoffman-Singleton and Higman-Sims. The electronic journal of combinatorics, Tome 11 (2004) no. 1. doi: 10.37236/1830

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