On the Graphs of Hoffman-Singleton and Higman-Sims

  • Paul R. Hafner


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 $S(3,6,22)$.

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