## On the Terwilliger algebra of bipartite distance-regular graphs with $$\Delta_2 = 0$$ and $$c_2 = 2$$.(English)Zbl 1351.05066

Summary: Let $$\Gamma$$ denote a bipartite distance-regular graph with diameter $$D \geq 4$$ and valency $$k \geq 3$$. Let $$X$$ denote the vertex set of $$\Gamma$$, and let $$A$$ denote the adjacency matrix of $$\Gamma$$. For $$x \in X$$ and for $$0 \leq i \leq D$$, let $$\Gamma_i(x)$$ denote the set of vertices in $$X$$ that are distance $$i$$ from vertex $$x$$. Define a parameter $$\Delta_2$$ in terms of the intersection numbers by $$\Delta_2 = (k - 2)(c_3 - 1) -(c_2 - 1) p_{22}^2$$. It is known that $$\Delta_2 = 0$$ implies that $$D \leq 5$$ or $$c_2 \in \{1, 2 \}$$. For $$x \in X$$ let $$T = T(x)$$ denote the subalgebra of $$\mathrm{Mat}_X(\mathbb{C})$$ generated by $$A, E_0^\ast, E_1^\ast, \ldots, E_D^\ast$$, where for $$0 \leq i \leq D$$, $$E_i^\ast$$ represents the projection onto the $$i$$th subconstituent of $$\Gamma$$ with respect to $$x$$. We refer to $$T$$ as the Terwilliger algebra of $$\Gamma$$ with respect to $$x$$. By the endpoint of an irreducible $$T$$-module $$W$$ we mean $$\min\{i \mid E_i^\ast W \neq 0 \}$$. We find the structure of irreducible $$T$$-modules of endpoint 2 for graphs $$\Gamma$$ which have the property that for $$2 \leq i \leq D - 1$$, there exist complex scalars $$\alpha_i$$, $$\beta_i$$ such that for all $$x, y, z \in X$$ with $$\partial(x, y) = 2$$, $$\partial(x, z) = i$$, $$\partial(y, z) = i$$, we have $$\alpha_i + \beta_i | \Gamma_1(x) \cap \Gamma_1(y) \cap \Gamma_{i - 1}(z) | = | \Gamma_{i - 1}(x) \cap \Gamma_{i - 1}(y) \cap \Gamma_1(z) |$$, in case when $$\Delta_2 = 0$$ and $$c_2 = 2$$. The case when $$\Delta_2 = 0$$ and $$c_2 = 1$$ is already studied by M. S. MacLean et al. [Linear Algebra Appl. 496, 307–330 (2016; Zbl 1331.05237)]. We show that if $$\Gamma$$ 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, and we give the action of $$A$$ on this basis.

### MSC:

 05C12 Distance in graphs 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05E30 Association schemes, strongly regular graphs

Zbl 1331.05237
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### References:

 [1] Bannai, E.; Ito, T., (Algebraic Combinatorics I: Association schemes, Benjamin-Cummings Lecture Note, vol. 58, (1984), Menlo Park) [2] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-Regular Graphs, (1989), Springer-Verlag Berlin, Heidelberg · Zbl 0747.05073 [3] Caughman, J. S., The Terwilliger algebras of bipartite $$P$$- and $$Q$$-polynomial schemes, Discrete Math., 196, 65-95, (1999) · Zbl 0924.05067 [4] Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete Math., 187, 39-70, (1998) · Zbl 0958.05143 [5] Curtin, B., Bipartite distance-regular graphs, part I, Graphs Combin., 15, 143-158, (1999) · Zbl 0927.05083 [6] Curtin, B., Bipartite distance-regular graphs, part II, Graphs Combin., 15, 377-391, (1999) · Zbl 0939.05088 [7] Curtin, B., The local structure of a bipartite distance-regular graph, European J. Combin., 20, 739-758, (1999) · Zbl 0940.05074 [8] Curtin, B., Almost $$2$$-homogeneous bipartite distance-regular graphs, European J. Combin., 21, 865-876, (2000) · Zbl 1002.05069 [9] Curtis, C. W.; Reiner, I., Representation Theory of Finite Groups and Associative Algebras, (1962), AMS Chelsea Publishing · Zbl 0131.25601 [10] Go, J., The Terwilliger algebra of the hypercube, European J. Combin., 23, 399-429, (2002) · Zbl 0997.05097 [11] Hobart, S. A.; Ito, T., The structure of nonthin irreducible T-modules: ladder bases and classical parameters, J. Algebraic Combin., 7, 53-75, (1998) · Zbl 0911.05059 [12] Miklavič, Š., The Terwilliger algebra of a distance-regular graph of negative type, Linear Algebra Appl., 430, 251-270, (2009) · Zbl 1225.05257 [13] Miklavič, Š.; Penjić, S., On bipartite $$Q$$-polynomial distance-regular graphs with $$c_2 \leq 2$$, Electron. J. Combin., 21, 4, #P4.53, (2014) · Zbl 1305.05061 [14] M.S. MacLean, Š. Miklavič, On bipartite distance-regular graphs with exactly two irreducible $$T$$-modules with endpoint two, submitted for publication. [15] MacLean, M. S.; Miklavi č, Š.; Penjić, S., On the Terwilliger algebra of bipartite distance-regular graphs with $$\Delta_2 = 0$$ and $$c_2 = 1$$, Linear Algebra Appl., 496, 307-330, (2016) · Zbl 1331.05237 [16] Terwilliger, P., The subconstituent algebra of an association scheme (part I), J. Algebraic Combin., 1, 363-388, (1992) · Zbl 0785.05089 [17] Terwilliger, P., A new inequality for distance-regular graphs, Discrete Math., 137, 319-332, (1995) · Zbl 0814.05074
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