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Let $\Gamma$ denote a finite, simple and connected graph. Fix a vertex $x$ of$\Gamma$ which is not a leaf and let $T=T(x)$ denote the Terwilliger algebra of$\Gamma$ with respect to $x$. Assume that the unique irreducible $T$-modulewith endpoint $0$ is thin, or equivalently that $\Gamma$ ispseudo-distance-regular around $x$. We consider the property that $\Gamma$ has,up to isomorphism, a unique irreducible $T$-module with endpoint $1$, and thatthis $T$-module is thin. The main result of the paper is a combinatorialcharacterization of this property.

Certain graphs with exactly one irreducible $T$-module with endpoint $1$, which is thin: the pseudo-distance-regularized case