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

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Abstract

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.

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Acknowledgements

The author would like to thank Štefko Miklavič for helpful and constructive comments that greatly contributed to improving the final version of this article. This work is supported in part by the Slovenian Research Agency (research program P1-0285, research project J1-2451 and Young Researchers Grant).

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  1. Andrej Marušič Institute, University of Primorska, Muzejski trg 2, 6000, Koper, Slovenia

    Blas Fernández

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  1. Blas Fernández
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Correspondence to Blas Fernández.

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To the memory of Viviana P. Jaime.

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Fernández, B. Certain graphs with exactly one irreducible T-module with endpoint 1, which is thin. J Algebr Comb 56, 1287–1307 (2022). https://doi.org/10.1007/s10801-022-01155-w

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