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

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m(4) for every positive integer m. If m > 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m(4), 2m(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 4m(4) vertices. For odd m >= 3 the strongly regular graphs seem to be new. (C) 2009 Elsevier Ltd. All rights reserved.