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

Let Gamma denote a finite, simple, connected and bipartite 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. (C) 2021 Elsevier Ltd. All rights reserved.

Automated summary

This paper investigates the Terwilliger algebra of a finite, simple, connected, and bipartite graph, based on a specific vertex, and derives the property of the unique reducible T-module in the graph, providing a combinatorial characterization of this property.