Some constructions of PBIB designs

Summary Constructions of three series of regular GD and semi-regular GD designs are given. Furthermore, a series of rectangular PBIB designs is constructed and particular cases of this series which reduce to PBIB designs with two associate classes are also provided. Written while visiting Department of Statistics, University of Indore, India, December 1983 through February 1984.     

On spherical designs obtained from Q‐polynomial association schemes

We characterize that the image of the embedding of the Q ‐polynomial association scheme into the first eigenspace by primitive idempotent E ₁ is a spherical t‐design in terms of the Krein numbers. Furthermore, we show that the strengths of P‐ and Q‐polynomial schemes as spherical designs are bounded by a constant. Copyright © 2011 John Wiley & Sons, Ltd. 19:167‐177, 2011     

A construction of non-isomorphic amorphous association schemes from pseudo-cyclic association schemes

As a continuation of my paper [T. Fujisaki, A construction of amorphous association scheme from a pseudo-cyclic association scheme, Discrete Math. 285(1–3) (2004) 307–311], we show a construction of amorphous association scheme which is a fusion scheme of a direct product of two pseudo-cyclic association schemes with same first eigenmatrix. By using this construction, we can get at most three amorphous association scheme. We prove that if two pseudo-cyclic association scheme are non-isomorphic, then these three amorphous association schemes are mutually non-isomorphic.     

Investigations on association schemes with elementary abelian thin residue

In this paper, we give a new class of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of order $p^2$. We then study the automorphism groups of these schemes and determine whether these schemes are schurian.     

Some implications on amorphic association schemes

We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.     

On the connectedness of partially balanced block designs

The necessary and sufficient conditions for m-associate partially balanced block (PBB) designs to be connected are given. This generalizes the criterion for m-associate partially balanced incomplete block (PBIB) designs, which has originally been established by Ogawa, Ikeda and Kageyama (1984, Proceedings of the Seminar on Combinatorics and Applications, 248–255, Statistical Publishing Society, Calcutta).This work was partially supported by the Polish Academy of Sciences Grant No. MR I.1-2/2.     

A characterization of Q-polynomial association schemes

We prove a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial scheme.     

Four-class skew-symmetric association schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.     

Schur rings and non-symmetric association schemes on 64 vertices

In this paper we enumerate essentially all non-symmetric association schemes with three classes, less than 96 vertices and with a regular group of automorphisms. The enumeration is based on a computer search in Schur rings. The most interesting cases have 64 vertices.In one primitive case and in one imprimitive case where no association scheme was previously known we find several new association schemes. In one other imprimitive case with 64 vertices we find association schemes with an automorphism group of rank 4, which was previously assumed not to be possible.     

Commutativity of association schemes of prime square order having non-trivial thin closed subsets

Through a study of the structure of the modular adjacency algebra over a field of positive characteristic p for a scheme of prime order p and utilizing the fact that every scheme of prime order is commutative, we show that every association scheme of prime square order having a non-trivial thin closed subset is commutative. The second author was supported by Korea Research Foundation Grant (KRF-2006-003-00008).     

A category of association schemes

We define a category of association schemes and investigate its basic properties. We characterize monomorphisms and epimorphisms in our category. The category is not balanced. The category has kernels, cokernels, and epimorphic images. The category is not an exact category, but we consider exact sequences. Finally, we consider a full subcategory of our category and show that it is equivalent to the category of finite groups.     

On class dimension of flat association schemes in affine and affine-symplectic spaces

In this paper, we obtain upper bounds of the class dimension of flat association schemes in affine and affine-symplectic spaces and construct resolving sets for these schemes.     

A structural classification of SR group divisible designs

A simple method of construction of a semi-regular (SR) group divisible design from another SR group divisible design is given. Using this method, 111 available SR designs from Clatworthy (1973) and John and Turner (1977) are systematically classified into 20 classes. This procedure may produce new nonisomorphic solutions for known designs.     

Some constructions of two-associate class PBIB designs

Summary Two-associate class PBIB designs, having association schemes of GD orL ₂ types, are constructed by using patterned matrices and by methods of taking unions of sets of blocks.     

On non-symmetric commutative association schemes with exactly one pair of non-symmetric relations

For a given non-symmetric commutative association scheme, by fusing all the non-symmetric relations pairwise with their symmetric counterparts, we can obtain a new symmetric association scheme. In this paper, we introduce a set of feasibility and realizability conditions for a class e symmetric association scheme to be split into a class e+1 non-symmetric commutative association scheme. By applying the feasibility and realizability conditions, we obtain a classification into six categories of the class 4 non-symmetric fission schemes of group-divisible 3-schemes. Complete solutions for three of the six categories and partial results for the remaining cases are presented.     

Classification of skew‐Hadamard matrices of order 32 and association schemes of order 31

Using a backtracking algorithm along with an essential change to the rows of representatives of known 13 710 027 equivalence classes of Hadamard matrices of order 32, we make an exhaustive computer search feasible and show that there are exactly 6662 inequivalent skew‐Hadamard matrices of order 32. Two skew‐Hadamard matrices are considered SH ‐equivalent if they are similar by a signed permutation matrix. We determine that there are precisely 7227 skew‐Hadamard matrices of order 32 up to SH ‐equivalence. This partly settles a problem posed by Kim and Solé. As a consequence, we provide the classification of association schemes of order 31.     

Divisible design digraphs and association schemes

Divisible design digraphs are constructed from skew balanced generalized weighing matrices and generalized Hadamard matrices. Commutative and non-commutative association schemes are shown to be attached to the constructed divisible design digraphs.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

Amorphic association schemes with negative Latin square-type graphs

Applying results from partial difference sets, quadratic forms, and recent results of Brouwer and Van Dam, we construct the first known amorphic association scheme with negative Latin square-type graphs and whose underlying set is a nonelementary abelian 2-group. We give a simple proof of a result of Hamilton that generalizes Brouwer's result. We use multiple distinct quadratic forms to construct amorphic association schemes with a large number of classes.