Rows are connected to 16 other rows and to 6 designs == degree 22. Designs are connected to 14 rows and 8 disjoint designs == degree 22. Thus all 100 vertices have degree 22 each. Algebraic properties The automorphism group of the Higman–Sims graph is a group of order isomorphic to the semidirect product of the Higman–Sims group of order with the cyclic group of order 2. It has automorphisms that take any edge to any other edge, making the Higman–Sims graph an edge-transitive graph. The characteristic polynomial of the Higman–Sims graph is (x − 22)(x − 2)77(x + 8)22. Therefore, the Higman–Sims graph is an integral graph: its spectrum consists entirely of integers. It is also the only graph with this characteristic polynomial, making it a graph determined by its spectrum. Inside the Leech lattice 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. References Group theory Individual graphs Regular graphs Strongly regular graphs