Bordered Complex Hadamard Matrices and Strongly Regular Graphs

We consider bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph. Examples include a Butson-type complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph due to J. Wallis, and a family of Hadamard matrices given by Singh and Dubey. In this paper, we show that there is also a non Butson-type complex Hadamard matrix whose core is contained in the Bose-Mesner algebra of a conference graph, and prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose-Mesner algebra of a strongly regular graph.