Related papers: Character tables of association schemes based on a…
The bivariate $P$- and $Q$-polynomial structures of association schemes based on attenuated spaces are examined using recurrence and difference relations of the bivariate polynomials which form the eigenvalues of the scheme. These…
Association schemes are combinatorial objects that allow us solve problems in several branches of mathematics. They have been used in the study of permutation groups and graphs and also in the design of experiments, coding theory, partition…
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…
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,…
Fiol has characterized quotient-polynomial graphs as precisely the connected graphs whose adjacency matrix generates the adjacency algebra of a symmetric association scheme. We show that a collection of non-negative integer parameters of…
An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we study 4-class skew-symmetric association schemes. In J. Ma [On the nonexistence of skew-symmetric…
Leading towards the classification of primitive commutative association schemes as the ultimate goal, Bannai and some of his school have been trying to * identify the major sources of (primitive) commutative association schemes, * collect…
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…
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…
In this paper, the association scheme defined on the flags of a finite generalized quadrangle is considered. All possible fusions of this scheme are listed, and a full description for those of classes 2 and 3 is given. Furthermore, it is…
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…
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…
An imprimitive symmetric indecomposable association scheme of rank 5 is said to be Higmanian. A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a symmetric divisible design. We establish conditions which…
The study of $P$-polynomial association schemes (distance-regular graphs) and $Q$-polynomial association schemes, and in particular $P$- and $Q$-polynomial association schemes, has been a central theme not only in the theory of association…
In a labeling scheme the vertices of a given graph from a particular class are assigned short labels such that adjacency can be algorithmically determined from these labels. A representation of a graph from that class is given by the set of…
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…
Let $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ be the sets of quadratic forms and symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$, respectively. The orbits of $\mathscr{Q}(m,q)$ and $\mathscr{S}(m,q)$ under a…
In this paper we aim to characterize association schemes all of whose symmetric fusion schemes have only integral eigenvalues, and classify those obtained from a regular action of a finite group by taking its orbitals.
Bivariate P-polynomial association scheme of type $(\alpha,\beta)$ are defined as a generalization of the P-polynomial association schemes. This generalization is shown to be equivalent to a set of conditions on the intersection parameters.…
An association scheme is $P$-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate $P$-polynomial association schemes of type $(\alpha,\beta)$ were introduced by Bernard et al., and…