The Terwilliger algebra for the distance-regular graphs with valency three

Abstract

In this paper, we discuss a family of highly regular graphs, said to be distance-regular. We are particularly interested in the distance-regular graphs with valency three. It is known that there exist exactly 13 such graphs. Let $\hat{I}“$ denote a distance-regular graph with vertex set $X$. For any vertex $x\in X$, the corresponding Terwilliger algebra $T=T(x)$ is generated by the adjacency algebra $M$ of $\hat{I}“$ and the dual adjacency algebra $M^{\ast }=M^{\ast }(x)$ of $\hat{I}“$ with respect to $x$. It is known that the algebra $T$ is semisimple. By construction, the vector space $V={\mathbb{C}}^X$ is a module for $T$, said to be standard. In this paper we have the following goal. For each of the 13 distance-regular graphs $\hat{I}“$ with valency three, we will decompose the standard module $V$ into a direct sum of irreducible $T$-modules. Using this information, we will work out the dimension of $T$.

50 pages, 14 figures