The Terwilliger algebra $T$ has been extensively studied in the context of distance-regular graphs, which have only a few irreducible $T$-modules (up to isomorphism) of a specific endpoint, all of which are thin (with respect to a certain base vertex). This talk aims to extend these results to irreducible $T$-modules with endpoint 0 of certain (not necessarily distance-regular) graphs, and shed some new light on their combinatorial properties. We examine which vertices $x$ of a finite, simple, and connected graph $\Gamma$ admit a Terwilliger algebra $T=T(x)$ with an irreducible $T$-module with endpoint $0$, which is thin. We give a purely combinatorial characterization to this algebraic property, which involves the number of certain walks in $\Gamma$ of a specific shape.
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