Strongly regular graphs with parameters exist for all

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4 m 4 for every positive integer m . If m > 1 , such a Hadamard matrix is equivalent to a strongly regular graph with parameters ( 4 m 4 , 2 m 4 + m 2 , m 4 + m 2 , m 4 + m 2 ) . Strongly regular graphs with these parameters have been called max energy graphs, because they have maximal energy (as defined by Gutman) among all graphs on 4 m 4 vertices. For odd m ≥ 3 the strongly regular graphs seem to be new.