Strongly regular graphs with parameters (4m4,2m4+m2,m4+m2,m4+m2)(4m4,2m4+m2,m4+m2,m4+m2) exist for all m>1m>1

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m44m4 for every positive integer mm. If m>1m>1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m4,2m4+m2,m4+m2,m4+m2)(4m4,2m4+m2,m4+m2,m4+m2). 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 4m44m4 vertices. For odd m≥3m≥3 the strongly regular graphs seem to be new.