Let A be an association scheme on q ≥ 3 vertices. We show that the Bose-Mesner algebra of the generalized Hamming scheme H(n,A), for n ≥ 2, is not the Nomura algebra of a type II matrix. This result gives examples of formally self-dual Bose-Mesner algebras that are not the Nomura algebras of type II matrices.

We prove that there exist only "nitely many nontrivial graphical t-(v; k; ) designs when k 6 4t=3. This improves a previous result of Betten et al. (Discrete Math. 197/198 (1999) 83–109). c © 2001 Elsevier Science B.V. All rights reserved. We use the notation and terminology of [1] and assume that the reader is familiar with the concept of graphical t-designs [2]. All polynomials in this note a...

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 one...

A spin model is a square matrix W satisfying certain conditions which ensure that it yields an invariant of knots and links via a statistical mechanical construction of V. F. R. Jones. Recently F. Jaeger gave a topological construction for each spin model W of an association scheme which contains W in its Bose Mesner algebra. Shortly thereafter, K. Nomura gave a simple algebraic construction of...

We present a new approach to the construction of simple block designs. Using the computer package DISCRETA, we start with the construction of block designs which are invariant with respect to some prescribed group of automorphisms. Therefore, one applies the method of Kramer and Mesner which means that one has to solve systems of diophantine equations to get the designs. DISCRETA has proven its...

Type II matrices were introduced in connection with spin models for link invariants. It is known that a pair of Bose-Mesner algebras (called a dual pair) of commutative association schemes are naturally associated with each type II matrix. In this paper, we show that type II matrices whose Bose-Mesner algebras are imprimitive are expressed as so-called generalized tensor products of some type I...

A spin model is a square matrix that encodes the basic data for a statistical mechanical construction of link invariants due to V.F.R. Jones. Every spin model W is contained in a canonical Bose-Mesner algebra N (W ). In this paper we study the distance-regular graphs whose Bose-Mesner algebra M satisfies W ∈ M ⊆ N (W ). Suppose W has at least three distinct entries. We show that is 1-homogeneou...

A type II matrix is a square matrix W with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the const...

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

 We investigate the existence of some‎ ‎large sets of size nine‎. ‎The large set $LS[9](2,5,29)$ is constructed and‎ ‎existence of the family $LS[9](2,5,27l+j)$ for $lgeq 1, ‎2leq‎ ‎j