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Bipartite distance-regular graphs. I. (English) Zbl 0927.05083

Let \(Y=(X,\{R_i\}_{0\leq i\leq D})\) be a bipartite \(P\)- and \(Q\)-polynomial association scheme. Then \(A_0,\dots,A_D\) form a basis for the Bose-Mesner algebra \(M\) of \(Y\) (\(A_i\) is the \(i\)th associate matrix of \(Y\)). We now recall the dual Bose-Mesner algebra \(M^*\) of \(Y\). Fix any \(x\in X\). For integer \(i\) (\(0\leq i\leq D\)) let \(E_i^*=E_i^*(x)\) denote the diagonal matrix with the rows and columns indexed by \(X\) and \((E_i^*)yy=1\), if \(xy\in R_i\), and \((E_i^*)yy=0\), if \(xy\notin R_i\). We refer to \(E_i^*\) as the \(i\)th dual idempotent of \(Y\) with respect to \(x\). It follows that the matrices \(E_0^*,\dots,E_D^*\) form a basis for a algebra \(M^*=M^*(x)\) known as the dual Bose-Mesner algebra of \(Y\) with respect to \(x\). The subalgebra \(T=T(x)\) of \(\text{Mat}_X({\mathbb{C}})\) generated by \(M\) and \(M^*\) is called the Terwilliger algebra of \(Y\) with respect to \(x\). Let \(V\) denote the vector space \({\mathbb{C}}^X\) (column vectors). Then \(\text{Mat}_X({\mathbb{C}})\) acts on \({\mathbb{C}}^X\) by left multiplication. By a \(T\)-module, we mean a subspace \(W\) of \(V\) such that \(TW\subseteq W\). An irreducible \(T\)-module \(W\) is said to be thin whenever \(\dim E_i^*(W)\leq 1\) for \(0\leq i\leq D\). The endpoint of \(W\) is \(\min\{i\mid E_i^*(W)\neq 0\}\). Let \(\Gamma=(X,E)\) denote a bipartite distance-regular graph with diameter \(D\geq 4\) and fix a vertex \(x\) of \(\Gamma\). In this paper the structure of the (unique) irreducible \(T\)-module of endpoint \(0\) is determined. Up to isomorphism there is a unique irreducible \(T\)-module of endpoint \(1\). It is thin. Determined are the structure of each thin irreducible \(T\)-module \(W\) of endpoint \(1\) or \(2\) in terms of the intersection numbers of \(\Gamma\) and an additional real parameter \(\psi=\psi(W)\) in case the endpoint is \(2\), which the author refers to as the type of \(W\). It is assumed that each irreducible \(T\)-module of endpoint \(2\) is thin and the following two-fold result is obtained. First, the intersection numbers of \(\Gamma\) are determined by the diameter \(D\) of \(\Gamma\) and the set \(\{(\psi, \text{mult}(\psi))\mid \psi{\i}\Phi_2\}\), where \(\Phi_2\) denotes the set of distinct types of irreducible \(T\)-modules with endpoint \(2\), and where \(\text{mult}(\psi)\) denotes the the multiplicity with which the module of type \(\psi\) appears in the standard module \(V\). Secondly, the set \(\{(\psi,\text{mult}(\psi))\mid \psi{\i}\Phi_2\}\) is determined by the intersection numbers \(k\), \(b_2\), \(b_3\) of \(\Gamma\) and the spectrum of the graph \(\Gamma_2^2=\Gamma_2(\Gamma_2(x))\).

MSC:

05E30 Association schemes, strongly regular graphs
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