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Strongly regular graphs with parameters \((4m^{4},2m^{4}+m^{2},m^{4}+m^{2},m^{4}+m^{2})\) exist for all \(m>1\). (English) Zbl 1225.05252

Summary: 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 \(m\geq 3\) the strongly regular graphs seem to be new.

MSC:

05E30 Association schemes, strongly regular graphs
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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