On the graphs of Hoffman-Singleton and Higman-Sims

Abstract

We propose a new elementary definition of the Higman-Sims graph in which the 100 vertices are parametrised with ℤ4 × ℤ5 × ℤ5 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).