On the Trivial T-Module of a Graph
Let $\Gamma$ denote a finite, simple and connected graph. Fix a vertex $x$ of $\Gamma$ and let $T=T(x)$ denote the Terwilliger algebra of $\Gamma$ with respect to $x$. In this paper we study the unique irreducible $T$-module with endpoint $0$. We assume that this $T$-module is thin. The main result of the paper is a combinatorial characterization of this property.