## 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:

 5e+30 Association schemes, strongly regular graphs
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