Abstract We introduce a family of norms on the $n \times n$ complex matrices. These arise from probabilistic framework, and their construction validation involve probability theory, partition combinatorics, trace polynomials in noncommuting variables. As consequence, we obtain generalization Hunter’s positivity theorem for complete homogeneous symmetric polynomials.

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