REGULAR 2-GRAPHS ON 36 VERTICES

We describe an algorithm for the generation by computer of regular two -graphs, and using it we discover 136 new regular two-graphs on 36 ver tices, bringing the present known total of such regular two-graphs to 227. An analysis of the results obtained shows that many of these new regular two-graphs, and some of those that were already known, are rel ated and can be generated in a certain way from just three of them. Th e same algorithm was used to search for regular two-graphs on 30 verti ces and confirmed that the six found by Arlasarov in the mid 1970s, an d later by Bussemaker, Mathon, and Seidel, constitute the complete set . Each regular two-graph on 36 vertices gives rise to a Hadamard matri x of order 36 which may yield as many as 36(2) = 1296 pairwise nonisom orphic Hadamard 2-(35, 17, 8) designs. Using the computer we discovere d that the 227 regular two-graphs on 36 vertices determine 180 pairwis e nonisomorphic Hadamard matrices, and when these were analyzed for Ha damard designs, a total of 108,131 painwise nonisomorphic designs were found.