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.


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


Zbl 1331.05237
Full Text: DOI


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