Regular two-graphs on 36 vertices

Abstract

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 vertices, 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 related and can be generated in a certain way from just three of them. The same algorithm was used to search for regular two-graphs on 30 vertices and confirmed that the six found by Arlasarov in the mid 1970s, and later by Bussemaker, Mathon, and Seidel, constitute the complete set. Each regular two-graph on 36 vertices gives rise to a Hadamard matrix of order 36 which may yield as many as 36 2 = 1296 pairwise nonisomorphic Hadamard 2-(35, 17, 8) designs. Using the computer we discovered that the 227 regular two-graphs on 36 vertices determine 180 pairwise nonisomorphic Hadamard matrices, and when these were analyzed for Hadamard designs, a total of 108,131 pairwise nonisomorphic designs were found.