page_title: Higman–Sims graph
section_title: Inside the Leech lattice
hierarchical_section_title: Higman–Sims graph / Inside the Leech lattice
context_page_description: In mathematical graph theory, the Higman–Sims graph is a 22-regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg, i.e. no neighboring pair of vertices share a common neighbor and each non-neighboring pair of vertices share six common neighbors. It was first constructed by Mesner and rediscovered in 1968 by Donald G. Higman and Charles C. Sims as a way to define the Higman–Sims group, and that group is a subgroup of index two in the group of automorphisms of the Higman–Sims graph.
context_section_description: The Higman–Sims graph naturally occurs inside the Leech lattice: if X, Y and Z are three points in the Leech lattice such that the distances XY, XZ and YZ are  respectively, then there are exactly 100 Leech lattice points T such that all the distances XT, YT and ZT are equal to 2, and if we connect two such points T and T′ when the distance between them is , the resulting graph is isomorphic to the Higman–Sims graph.  Furthermore, the set of all automorphisms of the Leech lattice (that is, Euclidean congruences fixing it) which fix each of X, Y and Z is the Higman–Sims group (if we allow exchanging X and Y, the order 2 extension of all graph automorphisms is obtained).  This shows that the Higman–Sims group occurs inside the Conway groups Co₂ (with its order 2 extension) and Co₃, and consequently also Co₁.
caption: A projection of the Higman–Sims graph inside the Leech lattice.