Abstract. For positive integers m and n, we denote by BH(m, n) the set of all H ∈ M n×n (C) such that HH * = nI n and each entry of H is an m-th root of unity where H * is the adjoint matrix of H and I n is the identity matrix. For H 1 , H 2 ∈ BH(m, n) we say that H 1 is equivalent to H 2 if H 1 = P H 2 Q for some monomial matrices P, Q whose nonzero entries are m-th roots of unity. In this paper we classify BH(17, 17) up to equivalence by computer search.

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For a positive integer k we say that an association scheme with more than one point is k-equivalenced if each non-diagonal relation has valency k. In this paper we prove that every 3-equivalenced association scheme is Frobenius, that is, the set of relations coincides with the set of orbitals of a Frobenius group.

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In this paper, we give several simple criteria to detect possible periods and linking numbers for a given virtual link. We investigate the behavior of the generalized Alexander polynomial [Formula: see text] of a periodic virtual link [Formula: see text] via its Yang–Baxter state model given in [L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, in Diagrammatic Morphisms and Applications, Contemp. Math. 318 (2003) 113–140, arXiv:math/0112280v2 [math.GT] 31 Dec 2001].