Certain graphs with exactly one irreducible T-module with endpoint 1, which is thin

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. Assume that x is a distance-regularized vertex, which is not a leaf. We consider the property that Gamma has, up to isomorphism, a unique irreducible T-module with endpoint 1, and that this T-module is thin. The main result of the paper is a combinatorial characterization of this property.

Automated summary

The main result of this paper is a combinatorial characterization of a certain property of a graph Gamma, which has a unique irreducible T-module with endpoint 1, and that this T-module is thin.