### On a certain class of 1-thin distance-regular graphs

#### Abstract

Let *Γ* denote a non-bipartite distance-regular graph with vertex set *X*, diameter *D* ≥ 3, and valency *k* ≥ 3. Fix *x* ∈ *X* and let *T* = *T*(*x*) denote the Terwilliger algebra of *Γ* with respect to *x*. For any *z* ∈ *X* and for 0 ≤ *i* ≤ *D*, let *Γ*_{i}(*z*) = {*w* ∈ *X* : ∂(*z*, *w*) = *i*}. For *y* ∈ *Γ*_{1}(*x*), abbreviate *D*_{j}^{i} = *D*_{j}^{i}(*x*, *y*) = *Γ*_{i}(*x*) ∩ *Γ*_{j}(*y*) (0 ≤ *i*, *j* ≤ *D*). For 1 ≤ *i* ≤ *D* and for a given *y*, we define maps *H*_{i}: *D*_{i}^{i} → ℤ and *V*_{i}: *D*_{i − 1}^{i} ∪ *D*_{i}^{i − 1} → ℤ as follows: *H*_{i}(*z*) = |*Γ*_{1}(*z*) ∩ *D*_{i − 1}^{i − 1}|, *V*_{i}(*z*) = |*Γ*_{1}(*z*) ∩ *D*_{i − 1}^{i − 1}|.

We assume that for every *y* ∈ *Γ*_{1}(*x*) and for 2 ≤ *i* ≤ *D*, the corresponding maps *H*_{i} and *V*_{i} are constant, and that these constants do not depend on the choice of *y*. We further assume that the constant value of *H*_{i} is nonzero for 2 ≤ *i* ≤ *D*. We show that every irreducible *T*-module of endpoint 1 is thin. Furthermore, we show *Γ* has exactly three irreducible *T*-modules of endpoint 1, up to isomorphism, if and only if three certain combinatorial conditions hold. As examples, we show that the Johnson graphs *J*(*n*, *m*) where *n* ≥ 7, 3 ≤ *m* < *n*/2 satisfy all of these conditions.

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MANUSCRIPTDOI: https://doi.org/10.26493/1855-3974.2193.0b0

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications