Type-II Matrices Attached to Conference Graphs

We determine the Nomura algebras of the type-II matrices belonging to the Bose-Mesner algebra of a conference graph. 1 Type-II Matrices and Nomura Algebras We say that an n × n matrix W with complex entries is type II if W (j, i)(W)(i, j) = 1 n for i, j = 1, . . . , n. So a type-II matrix is invertible and has no zero entry. We use I and J to denote the identity matrix and the matrix of all ones respectively. For each integer n ≥ 2 and a complex number t satisfying nt + nt + 1 = 0, the matrix I + tJ is a type-II matrix. These matrices are known as the Potts models, which are examples of spin models. The matrices of spin models and four-weight spin models are also type II, see [1] and [8]. Type-II matrices arise from combinatorial objects. For instance, any Hadamard matrix is type II. Chris Godsil and the first author have shown that the Bose-Mesner algebra of any strongly regular graphs contains a typeII matrix that is not type-II equivalent to the Potts model, see [6]. Two type-II matrices W and W ′ are type-II equivalent if W ′ = M1WM2 for some monomial matrices M1 and M2. In [9], Jaeger, Matsumoto and Nomura have given the construction of a Bose-Mesner algebra from a type-II matrix. Let W be an n × n type-II