| We propose a new elementary definition of the Higman-Sims graph in which the 100 vertices are parametrised with $Z_4timesZ_5timesZ_5$ 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. |
Keywords
Hoffman-Singleton graph, Higman-Sims graph, Higman-Sims group, biaffine plane, S(3,6,22)
Math Review Classification
Primary 05C62, 05C25
; Secondary 05B25, 51E10, 51E26
Last Updated
19/12/03
Length
28 pages
Availability
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