Related papers: Association Scheme on Triples from the Unitary Gro…
Association schemes on triples (ASTs) are 3-dimensional analogues of classical association schemes. If a group acts two-transitively on a set, the orbits of the action induced on the triple Cartesian product of that set yields an AST. By…
Association schemes on triples (ASTs) are ternary analogues of classical association schemes. Analogous to Schurian association schemes, ASTs arise from the actions of two-transitive groups. In this paper, we obtain the sizes and third…
Association schemes on triples (ASTs) are ternary analogues of classical association schemes, whose relations and adjacency algebras are ternary instead of binary. We provide a survey of the current progress in the study of ASTs,…
In this paper, we obtain classification results for higher-dimensional analogues of classical association schemes called association schemes on triples (ASTs). We present an algorithm that enumerates all ASTs on a fixed number of vertices…
Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional…
An infinite family of association schemes obtained from the general unitary groups acting transitively on the sets of isotropic vectors in the finite unitary spaces are investigated. We compute the parameters and determine the character…
For any finite group $G$, and any positive integer $n$, we construct an association scheme which admits the diagonal group $D_n(G)$ as a group of automorphisms. The rank of the association scheme is the number of partitions of $n$ into at…
An association scheme is called partially metric if it has a connected relation whose distance-two relation is also a relation of the scheme. In this paper we determine the symmetric partially metric association schemes with a multiplicity…
We give three constructions of three-class association schemes as fusion schemes of the cyclotomic scheme, two of which are primitive
An association scheme is called amorphic if every possible fusion of relations gives rise to a fusion scheme. We call a pair of relations fusing if fusing that pair gives rise to a fusion scheme. We define the fusing-relations graph on the…
Alternative set theory (AST) may be suitable for the ones who try to capture objects or phenomenons with some kind of indefiniteness of a border. While AST provides various notions for advanced mathematical studies, correspondence of them…
Association schemes form one of the main objects of algebraic combinatorics, classically defined on finite sets. In this paper we define association schemes on arbitrary, possibly uncountable sets with a measure. We study operator…
In this paper, we give a characterization of the class of all circular-arc graphs whose schemes are association. Moreover, all association schemes which are the scheme of a circular-arc graph are characterized, specially it is proved that…
An association scheme is called amorphic if every possible fusion of relations gives rise to another association scheme. In earlier work, we showed that if an association scheme has at most one relation that is neither strongly regular of…
In this paper, we investigate hypergroups which arise from association schemes in a canonical way; this class of hypergroups is called realizable. We first study basic algebraic properties of realizable hypergroups. Then we prove that two…
The paper provides an introduction to the field of Algebraic Set Theory (AST). AST is a flexible categorical framework for studying different kinds of set theories: both classical and constructive, predicative and impredicative. We discuss…
We introduce quantum association schemes. This allows to define distance regular and strongly regular quantum graphs. We bring examples thereof. In addition, we formulate the duality for translation quantum association schemes corresponding…
A Tatra association scheme is an association scheme arising from a symmetric bilinear form defined on the equivalence classes of nonzero $2$-dimensional vectors modulo some subgroup of the multiplicative group of a finite field. In the…
Suppose that a binary operation $\circ$ on a finite set $X$ is injective in each variable separately and also associative. It is easy to prove that $(X,\circ)$ must be a group. In this paper we examine what happens if one knows only that a…
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…