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arXiv · 2606.05393

The Terwilliger algebra for the distance-regular graphs with valency three

Kevin Kauflin·Paul Terwilliger·Barnabás Valkó·Jimmy Vineyard·Hanyi Wu
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Abstract

In this paper, we discuss a family of highly regular graphs, said to be distance-regular. We are particularly interested in the distance-regular graphs with valency three. It is known that there exist exactly 13 such graphs. Let $\Gamma$ denote a distance-regular graph with vertex set $X$. For any vertex $x \in X$, the corresponding Terwilliger algebra $T=T(x)$ is generated by the adjacency algebra $M$ of $\Gamma$ and the dual adjacency algebra $M^*=M^*(x)$ of $\Gamma$ with respect to $x$. It is known that the algebra $T$ is semisimple. By construction, the vector space $V=\mathbb{C}^X$ is a module for $T$, said to be standard. In this paper we have the following goal. For each of the 13 distance-regular graphs $\Gamma$ with valency three, we will decompose the standard module $V$ into a direct sum of irreducible $T$-modules. Using this information, we will work out the dimension of $T$.

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Kevin Kauflin, Paul Terwilliger, Barnabás Valkó, Jimmy Vineyard, Hanyi Wu. 2026-06-03. The Terwilliger algebra for the distance-regular graphs with valency three. https://arxiv.org/abs/2606.05393

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