Non-symmetric Association Schemes Research Articles

The search for small association schemes with noncyclotomic eigenvalues

In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative association scheme corresponds to a standard integral table algebra with integral multiplicities that satisfies all of the parameter restrictions known to hold for association schemes. For each rank and involution type, we generate an algebraic set for which any suitable integral solution corresponds to a standard integral table algebra with integral multiplicities, and then try to find the smallest suitable solution. The main results of this paper show the eigenvalues of association schemes of rank 4 and nonsymmetric association schemes of rank 5 will always be cyclotomic. In the rank 5 cases, the results rely on calculations done by computer for Gröbner bases or for bases of rational vector spaces spanned by polynomials. We give several examples of feasible parameter sets for small symmetric association schemes of rank 5 that have noncyclotomic eigenvalues.

A construction of pairs of non-commutative rank 8 association schemes from non-symmetric rank 3 association schemes

We construct a pair of non-commutative rank 8 association schemes from a rank 3 non-symmetric association scheme. For the pair, two association schemes have the same character table but different Frobenius-Schur indicators. This situation is similar to the pair of the dihedral group and the quaternion group of order 8. We also determine the structures of adjacency algebras of them over the rational number field.

Complex Hadamard Matrices Attached to a 3-Class Nonsymmetric Association Scheme

In this paper we classify complex Hadamard matrices contained in the Bose-Mesner algebra of nonsymmetric 3-class association schemes. As a consequence of our classification, we have two infinite families and some small examples of complex Hadamard matrices contained in the Bose-Mesner algebra of a self-dual fission of a complete multipartite graph.

Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

Abstract We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

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.

Distance Biregular Bipartite Graphs

We describe here some properties of a class of graphs which extends the class of distance regular graphs: our graphs are bipartite and for each vertex there exists an intersection array depending on the stable component of the vertex. Thus our graphs are to distance regular graphs as bipartite regular graphs are to regular graphs. They also are to non-symmetric association schemes as distance regular graphs are to symmetric association schemes. We also give some examples.

On splitting the Clebsch graph

The Clebsch graph can, as any strongly regular graph, be considered as one of the graphs of a symmetric association scheme with two classes. In this paper we show that this scheme is not the closure of any non-symmetric association scheme with three classes. This implies that no primitive non-symmetric association schemes with three classes on 16 elements exist.

Non-symmetric association schemes of symmetric matrices

LetXn be the set ofn ×n symmetric matrices over a finite fieldFq, whereq is a power of an odd prime. ForS1,S2 eXn, we define (S1,S2) eR0 iffS1 =S2; (S1,S2) eR(r,ξ) iffS1 -S2 is congruent to $$\left( {\begin{array}{*{20}c} {I^{(r - 1)} } & {} & {} \\ {} & \xi & {} \\ {} & {} & {0^{(n - r)} } \\ \end{array} } \right),$$ where ξ = 1 orz,z being a fixed non-square element ofFq. ThenXn = (Xn, {R0,R(r,ξ) | 1 ≤r ≤n, ξ = 1 orz}) is a non-symmetric association scheme of class 2n onXn. The parameters ofXn have been computed. And we also prove thatXn is commutative.