One finds a relationship between the theory of unimodular lattices and the theory of strongly regular graphs. For a unimodular, even lattice of dimension 32 having a system of roots of type A1 one constructs a strongly regular graph with parameters n=8184, a=7595, c=7042, d=7130. The graphs that arise from certain “Steiner sixtuple systems” have the same parameters. One also constructs strongly regular graphs for extremal lattices of dimension 48.