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

DOI10.1007/S10801-022-01155-WarXiv2301.10143MaRDI QIDQ6424292

Author name not available (Why is that?)

Publication date: 24 January 2023

Abstract: 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 $T$-module with endpoint $0$ is thin, or equivalently that $Gamma$ is pseudo-distance-regular around $x$. 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.

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