# On the Terwilliger algebra of certain family of bipartite distance-regular graphs with $\Delta_2=0$

### Abstract

Let *Γ* denote a bipartite distance-regular graph with diameter *D* ≥ 4 and valency *k* ≥ 3. Let *X* denote the vertex set of *Γ*, and let *A*_{i} (0 ≤ *i* ≤ *D*) denote the distance matrices of *Γ*. We abbreviate *A* := *A*_{1}. For *x* ∈ *X* and for 0 ≤ *i* ≤ *D*, let *Γ*_{i}(*x*) denote the set of vertices in *X* that are distance *i* from vertex *x*.

Fix *x* ∈ *X* and let *T* = *T*(*x*) denote the subalgebra of Mat_{X}(ℂ) generated by *A*, *E*_{0}^{*}, *E*_{1}^{*}, …, *E*_{D}^{*}, where for 0 ≤ *i* ≤ *D*, *E*_{i}^{*} represents the projection onto the *i*th subconstituent of *Γ* with respect to *x*. We refer to *T* as the *Terwilliger algebra* of *Γ* with respect to *x*. By the *endpoint* of an irreducible *T*-module *W* we mean min{*i* ∣ *E*_{i}^{*}*W* ≠ 0}.

In this paper we assume *Γ* has the property that for 2 ≤ *i* ≤ *D* − 1, there exist complex scalars *α*_{i}, *β*_{i} such that for all *y*, *z* ∈ *X* with ∂(*x*, *y*) = 2, ∂(*x*, *z*) = *i*, ∂(*y*, *z*) = *i*, we have *α*_{i} + *β*_{i}|*Γ*_{1}(*x*) ∩ *Γ*_{1}(*y*) ∩ *Γ*_{i − 1}(*z*)| = |*Γ*_{i − 1}(*x*) ∩ *Γ*_{i − 1}(*y*) ∩ *Γ*_{1}(*z*)|.

We study the structure of irreducible *T*-modules of endpoint 2. Let *W* denote an irreducible *T*-module with endpoint 2, and let *v* denote a nonzero vector in *E*_{2}^{*}*W*. We show that *W* = span({*E*_{i}^{*}*A*_{i − 2}*E*_{2}^{*}*v* ∣ 2 ≤ *i* ≤ *D*} ∪ {*E*_{i}^{*}*A*_{i + 2}*E*_{2}^{*}*v* ∣ 2 ≤ *i* ≤ *D* − 2}).

It turns out that, except for a particular family of bipartite distance-regular graphs with *D* = 5, this result is already known in the literature. Assume now that *Γ* is a member of this particular family of graphs. We show that if *Γ* is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible *T*-module with endpoint 2 and it is not thin. We give a basis for this *T*-module.