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

Abstract

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 subalgebra of Mat${}_X(\mathbb{C}$ generated by $A,E{\ast }_0,E\ast s_1,\dots ,E{\ast }_D$, where for $0\le i\le D$, $E{\ast }_i$ represents the projection onto the $i$th subconstituent of $\Gamma$ with respect to $x$. We refer to $T$ as the {\em Terwilliger algebra} of $\Gamma$ with respect to $x$. An irreducible $T$-module $W$ is said to be {\em thin} whenever dim $E{\ast }_{i}W\le 1$ for $0\le i\le D$. By the {\em endpoint} of $W$ we mean min$\{i|E{\ast }_{i}W\ne 0\}$. For $0\le i\le D$, let ${\Gamma }_i(z)$ denote the set of vertices in $X$ that are distance $i$ from vertex $z$. Define a parameter ${\Delta }_2$ in terms of the intersection numbers by ${\Delta }_2=(k-2)(c_3-1)-(c_2-1)p_{22}^2$. In this paper we prove the following are equivalent: (i) ${\Delta }_2>0$ and for $2\le i\le D-2$ there exist complex scalars ${\alpha }_i,{\beta }_i$ with the following property: for all $x,y,z\in X$ such that $\partial (x,y)=2,\partial (x,z)=i,\partial (y,z)=i$ we have ${\alpha }_i+{\beta }_i|{\Gamma }_1(x)\cap {\Gamma }_1(y)\cap {\Gamma }_{i-1}(z)|=|{\Gamma }_{i-1}(x)\cap {\Gamma }_{i-1}(y)\cap {\Gamma }_1(z)|;$ (ii) For all $x\in X$ there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra $T(x)$ with endpoint two, and these modules are thin.

References in corpus (1)

• Taut distance-regular graphs and the subconstituent algebra