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 4m4 for every positive integerm. Ifm > 1, such a Hadamard matrix is equivalent to a strongly regular graph with parameters (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 4m4 vertices. For oddm ≥ 3 the strongly regular graphs seem to be new. © 2009 Elsevier Ltd. All rights reserved. 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 r + s = λ− μ, rs = μ− k, f + g = n− 1, k+ fr + gs = 0. (1) 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 graphwas introduced by Gutman in 1978 (see [5]), and it originated from theoretical chemistry. The recent talk by Stevanović [10] provides a good survey of research results on energy of graphs. E-mail addresses: [email protected] (W.H. Haemers), [email protected] (Q. Xiang). 0195-6698/$ – see front matter© 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ejc.2009.07.009 1554 W.H. Haemers, Q. Xiang / European Journal of Combinatorics 31 (2010) 1553–1559 If Γ is an srg, E(Γ ) = k + fr − gs = −2gs. By the use of (1) it is an easy exercise to see that the srg’s of the title have energy 2m4(1+ 2m2). 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 ( n, n+ √ n 2 , n+ 2 √ n 4 , n+ 2 √ n 4 )