Strongly Regular Graphs with Parameters ( 4 m 4 , 2 m 4 + m 2 , m 4 + m 2 , m 4 + m 2 ) Exist for

Using results on Hadamard difference sets, we construct regular graphical Hadamard matrices of negative type of order 4m for every positive integer m. If m > 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (4m, 2m +m, m +m, m +m). 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 vertices. For odd m ≥ 3 the strongly regular graphs seem to be new. 1. Max energy graphs A strongly regular graph (srg) with parameters (n, k, λ, μ) is a graph with n vertices that is regular of valency k (1 ≤ k ≤ n − 2) and that has the following properties: • For any two adjacent vertices x, y, there are exactly λ vertices adjacent to both x and y. • For any two nonadjacent vertices x, y, there are exactly μ vertices adjacent to both x and y. A disconnected srg is the disjoint union of cliques of the same size. The adjacency matrix of a connected srg with parameters (n, k, λ, μ) has three distinct eigenvalues k, r and s (k > r ≥ 0 > s), of multiplicity 1, f and g, respectively, where (1) r + s = λ− μ, rs = μ− k, f + g = n− 1, k + fr + gs = 0. The energy E(Γ) of a graph Γ is the sum of the absolute values of the eigenvalues of its adjacency matrix. The concept of energy of a graph was introduced by Gutman in 1978 (see [5]), and it originated from theoretical chemistry. The recent talk by Stevanović [11] provides a good survey of research results on energy of graphs. If Γ is an srg, E(Γ) = k + fr − gs = −2gs. By use of (1) it is an easy exercise to see that the srg’s of the title have energy 2m(1 + 2m). This equals an upper bound on the energy by Koolen and Moulton [8], who proved the following result. Theorem 1. Let Γ be a graph on n vertices. Then E(Γ) ≤ n(1 + √ n) 2 , with equality holding if and only if Γ is an srg with parameters (2) (n, n+ √ n 2 , n+ 2 √ n 4 , n+ 2 √ n 4 ).