Hoffman–Singleton graph

The graph on the 15-cocliques, adjacent when they meet in 8 points, is the unique distance-regular graph with intersection array {15,14,10,3;1,5,12,15}. It has full group PSU(3,5).2 with point stabilizer A7 and edge stabilizer L2(7):2 (see below). c) Split into two copies of 5C5 . There are 126 of these, forming a single orbit. (The construction given above gives an explicit split.) The stabilizer is 5+1+2:8:2 with vertex orbit size 50. The subgraph induced on the orbit of size 36 is the Sylvester graph , the unique distance-regular graph with intersection array {5,4,2;1,1,4}. Each Petersen graph is split 5+5 in 6 splits into two 5C5. Each split into two 5C5 determines 25 Petersen graphs. Each pair of splits determines a unique Petersen graph. In this way we find the unital in PG(2,52), with splits into two 5C5 as points, and Petersen graphs as lines. The Hoffman-Singleton graph has independence number 15, and chromatic number 4. It has edge-chromatic number 7. The complement of the Hoffman-Singleton graph has independence number 2, and chromatic number 25. W.H. Haemers, A new partial geometry constructed from the Hoffman-Singleton graph , Finite Geometries and designs, Proc. Second Isle of Thorns Conference 1980, P.J. Cameron, J.W.P. Hirschfeld & D.R. Hughes (eds.), London Math. Soc. Lecture Note Ser. 49 , Cambridge University Press, Cambridge (1981) 119-127.