Taut distance-regular graphs and the subconstituent algebra

A diagram associated with the subconstituent algebra of a distance-regular graph

In this paper we consider a distance-regular graph $Γ$. Fix a vertex $x$ of $Γ$ and consider the corresponding subconstituent algebra $T$. The algebra $T$ is the $\mathbb{C}$-algebra generated by the Bose-Mesner algebra $M$ of $Γ$ and the dual Bose-Mesner algebra $M^*$ of $Γ$ with respect to $x$. We consider the subspaces $M, M^*, MM^*, M^*M, MM^*M, M^*MM^*, \dots$ along with their intersections and sums. In our notation, $MM^*$ means $Span\{RS|R\in M, S\in M^*\}$, and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram u…

Distance-regular graphs of $q$-Racah type and the $q$-tetrahedron algebra

In this paper we discuss a relationship between the following two algebras: (i) the subconstituent algebra $T$ of a distance-regular graph that has $q$-Racah type; (ii) the $q$-tetrahedron algebra $\boxtimes_q$ which is a $q$-deformation of the three-point $sl_2$ loop algebra. Assuming that every irreducible $T$-module is thin, we display an algebra homomorphism from $\boxtimes_q$ into $T$ and show that $T$ is generated by the image together with the center $Z(T)$.

Strongly Regular Graphs with No Triangles

A simplified version of the theory of strongly regular graphs is developed for the case in which the graphs have no triangles. This leads to (i) direct proofs of the Krein conditions, and (ii) the characterization of strongly regular graphs with no triangles such that the second subconstituent is also strongly regular. The method also provides an effective means of listing feasible parameters for such graphs.

A group commutator involving the last distance matrix and dual distance matrix of a $Q$-polynomial distance-regular graph

Let $Γ$ denote the Hamming graph $H(D,r)$ with $r \geq 3$. Consider the distance matrices $\{A_i\}_{i=0}^{D}$ of $Γ$. Fix a vertex $x$ of $Γ$, and consider the dual distance matrices $\{A_i^{*}\}_{i=0}^{D}$ of $Γ$ with respect to $x$. We investigate the group commutator $A_{D}^{-1}A_{D}^{*-1}A_{D}A_{D}^{*}$. We show that this matrix is diagonalizable. We compute its eigenvalues and their eigenspaces. Let $T$ denote the subconstituent algebra of $Γ$ with respect to $x$. We describe the action of $A_{D}^{-1}A_{D}^{*-1}A_{D}A_{D}^{*}$ on each irreducible $T$-module.

Orthogonal graphs modulo power of 2

In this work, we define an orthogonal graph on the set of equivalence classes of $(2ν+ δ)-$tuples over $\mathbb{Z}_{2^n}$ where $n$ and $ν$ are positive integers and $δ= 0, 1$ or $2$. We classify our graph if it is strongly regular or quasi-strongly regular and compute all parameters precisely. We show that our graph is arc transitive. The automorphisms group is given and the chromatic number of the graph except when $δ= 0$ and $ν$ is odd is determined. Moreover, we work on subconstituents of this orthogonal graph.

伪几何强正则图与一个规则点

Pseudo-Geometric Strongly Regular Graphs with a Regular Point

本文研究了一类伪几何强正则图，其第二子构成是某个强正则图或完全图的覆盖。通过分析此类图的结构，作者找到了许多新的强正则图，并回答了Gardiner等人提出的问题。

自旋模型与$q$-Racah型距离正则图

Spin models and distance-regular graphs of $q$-Racah type

本文研究形式自对偶且$q$-Racah型的距离正则图，利用其子约束代数中的中心元素构造自旋模型，并探讨该元素的组合含义及从自旋模型恢复该元素的方法，最后提出开放问题。

Distance-regular graphs of $q$-Racah type and the universal Askey-Wilson algebra

Let $\C$ denote the field of complex numbers, and fix a nonzero $q \in \C$ such that $q^4 \ne 1$. Define a $\C$-algebra $Δ_q$ by generators and relations in the following way. The generators are $A,B,C$. The relations assert that each of $A+\frac{qBC-q^{-1}CB}{q^2-q^{-2}}$, $B+\frac{qCA-q^{-1}AC}{q^2-q^{-2}}$, $C+\frac{qAB-q^{-1}BA}{q^2-q^{-2}}$ is central in $Δ_q$. The algebra $Δ_q$ is called the universal Askey-Wilson algebra. Let $Γ$ denote a distance-regular graph that has $q$-Racah type. Fix a vertex $x$ of $Γ$ and let $T=T(x)$ denote the corresponding subconstituent algebra. In this pape…

$Q$-多项式距离正则图的核

The nucleus of a $Q$-polynomial distance-regular graph

本文研究了$Q$-多项式距离正则图的核结构，探讨了核的多重视角，并在非二部双极图假设下给出了核的基底和作用方式。

从距离正则图中构造关系代数表示

Relation Algebra Representations from Distance-Regular Graphs

本文提出通过距离正则图构造有限整数对称关系代数表示的方法，证明了当图也是距离传递时表示为代数的充要条件，并展示了直径为3时的特殊情况及应用。

双线性形式图的距离正则图的公平划分

An equitable partition for the distance-regular graph of the bilinear forms

本文研究双线性形式图的距离正则图的公平划分，提出(x,y)-划分，并将其特征向量作为T-模块的基础，分解为不可约T-模块的正交直和。

On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint 2

Let $Γ$ denote a bipartite distance-regular graph with diameter $D \ge 4$ and valency $k \ge 3$. Let $X$ denote the vertex set of $Γ$, and let $A$ denote the adjacency matrix of $Γ$. For $x \in X$ let $T=T(x)$ denote the subalgebra of Mat$_X(\mathbb{C}$ generated by $A, E*_0, E*s_1, \ldots, E*_D$, where for $0 \le i \le D$, $E*_i$ represents the projection onto the $i$th subconstituent of $Γ$ with respect to $x$. We refer to $T$ as the {\em Terwilliger algebra} of $Γ$ with respect to $x$. An irreducible $T$-module $W$ is said to be {\em thin} whenever dim $E*_i W \le 1$ for $0 \le i \le D$. By…

AT4族的一个新可行性条件

A New Feasibility Condition for the AT4 Family

本文研究AT4(p,q,r)对跖紧距离正则图，通过第二子构成的对跖紧性给出新可行性条件，并据此排除一类参数、讨论特殊r情形。

关于小直径伪顶点传递图的注记

Remarks on pseudo-vertex-transitive graphs with small diameter

本文研究直径为2、3、4的距离正则图的伪顶点传递性，分别刻画了强正则图、Taylor图和antipodal tight图的伪顶点传递条件。