Two-graphs on 36 Points from the Even Unimodular Lattices E 8 E 8 and D + 16

We show in this paper a power of a construction 5], 6] of two-graphs from doubly even lattices. A doubly even lattice is an even lattice multiplied by p 2. We apply this construction to the even unimodular lattices E 8 E 8 and D + 16 multiplied by p 2. For the lattice p 2(E 8 E 8), the construction gives one family of regular two-graphs on 36 points. We think that many of two-graphs of this family are new. The lattice p 2D + 16 provides two families of such two-graphs. One of these families from D + 16 consists of all two-graphs related to Steiner triple systems on 15 points. Another family consists of two-graphs related to 2-(10,4,2) designs. This family was discovered by T.Spence, by use a computer. These two-graphs are brieey described by J.Seidel in 10]. Being distinct all the 3 families are not disjoint. In particular, all contain the 2 two-graph related to the Steiner triple systems having No 1 and No 2 in the extended version of 2]. We do not consider here the hard problem on a number of isomorphism classes of two-graphs in each family, but show that root systems related to two-graphs make possible to distinguish nonisomorphic two-graphs.