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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
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[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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