A connection between Hadamard matrices, oriented hypergraphs and signed graphs

Matrices associated to oriented hypergraphs produce a connection between signed graphs and Hadamard matrices. The existence of a family of signed graphs that are switching equivalent to −Kn and whose adjacency matrices sum to the zero matrix is shown to be equivalent to the existence of a Hadamard matrix. This equivalent problem is used to make explicit signed graph constructions which specialize to known Hadamard constructions.