Uploaded October 2010 | Updated September 2026, 2 weeks ago
The Higman-Sims graph is the unique graph with 100 vertices such that each is adjacent to 22 others and no two adjacent vertices have a common neighbor (i.e., the graph has no triangle) and any two non-adjacent vertices have exactly six common neighbors. It has 88704000 automorphism, forming an extension of 2 by the unique simple group of order 44352000 (the Higman-Sims group, a sporadic group).
The Higman-Sims graph occurs inside the 24-dimensional Leech lattice (if X,Y,Z are Leech lattice points at distances 3,3,2 from each other, then there are 100 Leech lattice points at distance 2,2,2 from X,Y,Z, and if we connect those at distance 3 from another, we obtain the H-S graph).
This animation displays various orthogonal projections of the H-S graph inside the Leech lattice, chosen so as to reveal an 11-fold symmetry (there is only one conjugacy class of order 11 in ·0, which is in HS).
The Higman-Sims graph is the unique graph with 100 vertices such that each is adjacent to 22 others and no two adjacent vertices have a common neighbor (i.e., the graph has no triangle) and any two non-adjacent vertices have exactly six common neighbors. It has 88704000 automorphism, forming an extension of 2 by the unique simple group of order 44352000 (the Higman-Sims group, a sporadic group).
The Higman-Sims graph occurs inside the 24-dimensional Leech lattice (if X,Y,Z are Leech lattice points at distances 3,3,2 from each other, then there are 100 Leech lattice points at distance 2,2,2 from X,Y,Z, and if we connect those at distance 3 from another, we obtain the H-S graph).
This animation displays various orthogonal projections of the H-S graph inside the Leech lattice, chosen so as to reveal an 11-fold symmetry (there is only one conjugacy class of order 11 in ·0, which is in HS).
