相关论文: The Terwilliger Algebra of a Distance-Regular Grap…

Let $\Gamma$ denote a bipartite distance-regular graph with diameter $D \ge 4$ and valency $k \ge 3$. Let $X$ denote the vertex set of $\Gamma$, and let $A$ denote the adjacency matrix of $\Gamma$. For $x \in X$ let $T=T(x)$ denote the…

Let $\Gamma$ denote a finite, simple and connected graph. Fix a vertex $x$ of $\Gamma$ which is not a leaf and let $T=T(x)$ denote the Terwilliger algebra of $\Gamma$ with respect to $x$. Assume that the unique irreducible $T$-module with…

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3$ and Bose-Mesner algebra $M$. For $\theta\in C\cup \infty$ we define a 1 dimensional subspace of $M$ which we call $M(\theta)$. If $\theta\in C$ then $M(\theta)$ consists…

Let Gamma be a Q-polynomial distance-regular graph with vertex set X, diameter D geq 3 and adjacency matrix A. Fix x in X and let A*=A*(x) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T=T(x) is the…

Let $Y$ denote a $D$-class symmetric association scheme with $D \geq 3$, and suppose $Y$ is almost-bipartite P- and Q-polynomial. Let $x$ denote a vertex of $Y$ and let $T=T(x)$ denote the corresponding Terwilliger algebra. We prove that…

The Terwilliger algebra $T(x)$ of a finite connected simple graph $\Gamma$ with respect to a vertex $x$ is the complex semisimple matrix algebra generated by the adjacency matrix $A$ of $\Gamma$ and the diagonal matrices…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D\geq 1$. For a vertex $x$ of $\Gamma$ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $\Gamma$ and the dual adjacency…

Let $\Gamma$ be a $Q$-polynomial distance-regular graph with diameter at least $3$. Terwilliger (1993) implicitly showed that there exists a polynomial, say $T(\lambda)\in \mathbb{C}[\lambda]$, of degree $4$ depending only on the…

In the year 2000, Eric Egge introduced the generalized Terwilliger algebra $\mathcal T$ of a distance-regular graph $\Gamma$. For any vertex $x$ of $\Gamma,$ there is a surjective algebra homomorphism $\natural$ from $\mathcal T$ to the…

Let $\Gamma=(X,\mathcal{R})$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set $X$ and edge set $\mathcal{R}$. Fix a vertex $x \in X$, and define $\mathcal{R}_f = \mathcal{R} \setminus \{yz \mid…

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$ and $b\not=1$, $\alpha=b-1$. The condition on $\alpha$ implies that $\Gamma$ is formally self-dual. For $b=q^2$ we use the adjacency matrix and dual…

Let $\Gamma^{(x_0)}$ be a finite rooted tree, for which $\Gamma$ is the underlying tree and $x_0$ the root. Let $T$ be the Terwilliger algebra of $\Gamma$ with respect to $x_0$. We study the structure of the principal $T$-module. As a…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. For a vertex $x\in X$ and for $0 \leq i \leq D$, let $E^*_i(x)$ denote the projection matrix…

Let $D\geq 3$ denote an integer. For any $x\in \mathbb F_2^D$ let $w(x)$ denote the Hamming weight of $x$. Let $X$ denote the subspace of $\mathbb F_2^D$ consisting of all $x\in \mathbb F_2^D$ with even $w(x)$. The $D$-dimensional halved…

Let $\Gamma$ be a dual polar graph with diameter $D \geqslant 3$, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field $\mathbb{F}_q$ equipped with a non-degenerate form (alternating,…

Let $\Gamma$ denote a distance-regular graph with vertex set $X$ and diameter $D \geq 3$. Fix a vertex $x \in X$. Let the field $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$. Let $\operatorname{Mat}_X(\mathbb{F})$ denote the…

Let $\Gamma = (\Omega,E)$ be a strongly-regular graph with adjacency matrix $A_1$, and let $A_2$ be the adjacency matrix of its complement. For any vertex $\omega\in \Omega$, we define $E_{0,\omega}^*$ $E_{1,\omega}^*$ and $E_{2,\omega}^*$…

For a finite connected simple graph, the Terwilliger algebra is a matrix algebra generated by the adjacency matrix and idempotents corresponding to the distance partition with respect to a fixed vertex. We will consider algebras defined by…

In this paper we consider a distance-regular graph $\Gamma$. Fix a vertex $x$ of $\Gamma$ and consider the corresponding subconstituent algebra $T$. The algebra $T$ is the $\mathbb{C}$-algebra generated by the Bose-Mesner algebra $M$ of…

We consider a bipartite distance-regular graph $G$ with diameter at least 4 and valency at least 3. Fix a vertex of $G$ and let $T$ denote the corresponding subconstituent algebra. We give a detailed description of a certain type of…