Divisible design digraphs and association schemes

每个非对称4类结合方案都可由一个有向图生成

Every nonsymmetric $4$-class association scheme can be generated by a digraph

该研究在特定假设下刻画了恰含一对非对称关系的交换结合方案的有向图生成条件，并应用该结论证明所有非对称4类结合方案都可由有向图生成，拓展了3类结合方案生成相关结论。

Factorisation of Greedoid Polynomials of Rooted Digraphs

Gordon and McMahon defined a two-variable greedoid polynomial $ f(G;t,z) $ for any greedoid $ G $. They studied greedoid polynomials for greedoids associated with rooted graphs and rooted digraphs. They proved that greedoid polynomials of rooted digraphs have the multiplicative direct sum property. In addition, these polynomials are divisible by $ 1 + z $ under certain conditions. We compute the greedoid polynomials for all rooted digraphs up to order six. A greedoid polynomial $ f(D) $ of a rooted digraph $ D $ of order $ n $ GM-factorises if $ f(D) = f(G) \cdot f(H) $ such that $ G $ and $ H…

Primitive weakly distance-regular circulant digraphs

We classify certain non-symmetric commutative association schemes. As an application, we determine all the primitive weakly distance-regular circulant digraphs.

On 3-equivalenced weakly distance-regular digraphs

A weakly distance-regular digraph is 3-equivalenced if its attached association scheme is 3-equivalenced. In this paper, we classify the family of such digraphs under the assumption of the commutativity.

The basis digraphs of p-schemes

It is proved that association schemes with bipartite basis graphs are exactly 2-schemes. This result follows from a characterization of p-schemes for an arbitrary prime p in terms of basis digraphs.

关于群可除结合方案直积的 Terwilliger $\mathbb{F}$-代数

On Terwilliger $\mathbb{F}$-algebras of direct products of group divisible association schemes

本文研究群可除结合方案直积的 Terwilliger $\mathbb{F}$-代数，获得了其中心、半单性、Jacobson 根及幂零指数和 Wedderburn-Artin 分解等结构性质。

Linked systems of symmetric group divisible designs

We introduce the concept of linked systems of symmetric group divisible designs. The connection with association schemes is established, and as a consequence we obtain an upper bound on the number of symmetric group divisible designs which are linked. Several examples of linked systems of symmetric group divisible designs are provided.

Linked systems of symmetric group divisible designs of type II

The linked systems of symmetric group divisible designs of type II is introduced, and several examples are obtained from affine resolvable designs and mutually UFS Latin squares. Furthermore, an equivalence between such symmetric group divisible designs and some association schemes with $5$-classes is provided.

A duality of scaffolds for translation association schemes

Scaffolds are certain tensors arising in the study of association schemes, and have been (implicitly) understood diagrammatically as digraphs with distinguished "root" nodes and with matrix edge weights, often taken from Bose-Mesner algebras. In this paper, we first present a slight modification of Martin's conjecture (2021) concerning a duality of scaffolds whose digraphs are embedded in a closed disk in the plane with root nodes all lying on the boundary circle, and then show that this modified conjecture holds true if we restrict ourselves to the class of translation association schemes, i.…

Commutative association schemes obtained from twin prime powers, Fermat primes, Mersenne primes

For prime powers $q$ and $q+\varepsilon$ where $\varepsilon\in\{1,2\}$, an affine resolvable design from $\mathbb{F}_q$ and Latin squares from $\mathbb{F}_{q+\varepsilon}$ yield a set of symmetric designs if $\varepsilon=2$ and a set of symmetric group divisible designs if $\varepsilon=1$. We show that these designs derive commutative association schemes, and determine their eigenmatrices.

Almost Difference Sets, Normally Regular Digraphs and Cyclotomic Schemes from Cyclotomy of Order Twelve

Using cyclotomic classes of order twelve for certain finite fields, we construct an infinite family of almost difference sets and normally regular graphs applying the theory of cyclotomy. We show that in each of these fields neither the multiplicative cyclic subgroup $C$ of index twelve nor $C\cup \{0\}$ forms an almost difference set, but a union of cosets of $C$ provides us an almost difference set. We also calculate the intersection numbers and character tables of cyclotomic association schemes of class two, three and four obtained from these fields.

具有 $μ$ 接近半价数的充分正则图与群可分组设计

Amply regular graphs with $μ$ close to half the valency and group divisible designs

本文分类了直径 $d \geq 4$、参数满足 $μ = (k-1)/2$（$k \geq 5$ 为奇数）的连通充分正则图，证明它们只能是 5-立方体、$K_2 \square Λ$（$Λ$ 是 14 个顶点上的唯一二部 $(0,2)$-图），或具有对偶性质的群可分组设计的点-区关联图，并利用 Paley 图、Peisert 图和 Paley 有向图构造了无限族。

具有m-DCI性质的广义四元数群

Generalized quaternion groups with the m-DCI property

本文研究广义四元数群Q_{4n}的m-DCI性质，证明了当1≤m≤2n-1时n必为奇数且不被任何不超过m-1的素数的平方整除，并给出n为素数幂时的完整刻画。

由可分组设计构造的具有三个特征值的图

Graphs with three eigenvalues from group divisible designs

该研究从半正则可分组设计出发，利用Muzychuk的构造A、B及广义四边形等，构造出具有三个特征值的双正则图等，并确定其相干闭包的秩与类型，回答了相关问题。

Uniform Mixing and Association Schemes

We consider continuous-time quantum walks on distance-regular graphs of small diameter. Using results about the existence of complex Hadamard matrices in association schemes, we determine which of these graphs have quantum walks that admit uniform mixing. First we apply a result due to Chan to show that the only strongly regular graphs that admit instantaneous uniform mixing are the Paley graph of order nine and certain graphs corresponding to regular symmetric Hadamard matrices with constant diagonal. Next we prove that if uniform mixing occurs on a bipartite graph X with n vertices, then n i…