2004/11/03 by Paul R. Hafner · 1 citation
Engineering · Mathematics · Computer Science · #graph theory and CDMA systems #Finite Group Theory Research #Interconnection Networks and Systems #Singleton #Combinatorics #Mathematics #Transitive relation #Graph #Discrete mathematics
paper · pdf · doi:10.37236/1830
openalex publication_date 2004/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
We propose a new elementary definition of the Higman-Sims graph in which the 100 vertices are parametrised with \Bbb Z4×\Bbb Z5×\Bbb 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).