zbMATH — the first resource for mathematics

On the Terwilliger algebra of bipartite distance-regular graphs with $$\Delta_{2}=0$$ and $$c_{2}=1$$. (English) Zbl 1331.05237
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 $$\operatorname{\Gamma}_i(x)$$ denote the set of vertices in $$X$$ that are distance $$i$$ from vertex $$x$$. Define a parameter $$\operatorname{\Delta}_2$$ in terms of the intersection numbers by $$\operatorname{\Delta}_2 = (k - 2)(c_3 - 1) -(c_2 - 1) p_{22}^2$$. We first show that $$\operatorname{\Delta}_2 = 0$$ implies that $$D \leq 5$$ or $$c_2 \in \{1, 2 \}$$. For $$x \in X$$ let $$T = T(x)$$ denote the subalgebra of $$\text{Mat}_X(\mathbb{C})$$ 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 $$\min \{i | E_i^\ast W \neq 0 \}$$.
In this paper we assume $$\Gamma$$ has 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 $$\alpha_i + \beta_i | \operatorname{\Gamma}_1(x) \cap \operatorname{\Gamma}_1(y) \cap \operatorname{\Gamma}_{i - 1}(z) | = | \operatorname{\Gamma}_{i - 1}(x) \cap \operatorname{\Gamma}_{i - 1}(y) \cap \operatorname{\Gamma}_1(z) |$$. We additionally assume that $$\operatorname{\Delta}_2 = 0$$ with $$c_2 = 1$$.
Under the above assumptions we study the algebra $$T$$. We show that if $$\Gamma$$ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible $$T$$-module with endpoint 2. We give an orthogonal basis for this $$T$$-module, and we give the action of $$A$$ on this basis.

MSC:
 05E30 Association schemes, strongly regular graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C12 Distance in graphs
Full Text:
References:
  Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, Benjamin-Cummings Lecture Note Series, vol. 58, (1984), Benjamin-Cummings Menlo Park · Zbl 0555.05019  Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin · Zbl 0747.05073  Caughman, J. S., The Terwilliger algebras of bipartite P- and Q-polynomial schemes, Discrete Math., 196, 65-95, (1999) · Zbl 0924.05067  Curtin, B., Bipartite distance-regular graphs, part I, Graphs Combin., 15, 143-158, (1999) · Zbl 0927.05083  Curtin, B., Bipartite distance-regular graphs, part II, Graphs Combin., 15, 377-391, (1999) · Zbl 0939.05088  Curtin, B., 2-homogeneous bipartite distance-regular graphs, Discrete Math., 187, 39-70, (1998) · Zbl 0958.05143  Curtin, B., The local structure of a bipartite distance-regular graph, European J. Combin., 20, 739-758, (1999) · Zbl 0940.05074  Curtin, B., Almost 2-homogeneous bipartite distance-regular graphs, European J. Combin., 21, 865-876, (2000) · Zbl 1002.05069  Curtis, C. W.; Reiner, I., Representation theory of finite groups and associative algebras, (1962), AMS Chelsea Publishing · Zbl 0131.25601  Egge, E. S., A generalization of the Terwilliger algebra, J. Algebra, 233, 213-252, (2000) · Zbl 0960.05108  Go, J., The Terwilliger algebra of the hypercube, European J. Combin., 23, 399-429, (2002) · Zbl 0997.05097  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  M.S. MacLean, Š. Miklavič, On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two, submitted for publication.  Miklavič, Š., On bipartite Q-polynomial distance-regular graphs with $$c_2 = 1$$, Discrete Math., 307, 544-553, (2007) · Zbl 1112.05104  Miklavič, Š., On bipartite Q-polynomial distance-regular graphs, European J. Combin., 28, 94-110, (2007) · Zbl 1200.05262  Miklavič, Š., The Terwilliger algebra of a distance-regular graph of negative type, Linear Algebra Appl., 430, 251-270, (2009) · Zbl 1225.05257  Terwilliger, P., A new inequality for distance-regular graphs, Discrete Math., 137, 319-332, (1995) · Zbl 0814.05074  Terwilliger, P., The subconstituent algebra of an association scheme (part I), J. Algebraic Combin., 1, 363-388, (1992) · Zbl 0785.05089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.