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A new characterization of taut distance-regular graphs of odd diameter. (English) Zbl 1278.05251
Summary: We consider a bipartite distance-regular graph $$\varGamma$$ with vertex set $$X$$, diameter $$D\geq 4$$, and valency $$k\geq 3$$. Let $$\mathbb C^X$$ denote the vector space over $$\mathbb C$$ consisting of column vectors with rows indexed by $$X$$ and entries in $$\mathbb C$$. For $$z\in X$$, let $$\hat{z}$$ denote the vector in $$\mathbb C^X$$ with a 1 in the $$z^{th}$$ row and 0 in all other rows. For $$0\leq i\leq D$$, let $$\varGamma_i(z)$$ denote the set of vertices in $$X$$ that are distance $$i$$ from $$z$$. Fix $$x,y\in X$$ with distance $$\partial (x,y)=2$$. For $$0\leq i,j\leq D$$, we define $$w_{ij}=\sum\hat{z}$$, where the sum is over all vertices $$z\in\varGamma_i(x)\cap\varGamma_j(y)$$. Define a parameter $$\varDelta$$ in terms of the intersection numbers by $$\varDelta=(b_1-1)(c_3-1)-(c_2-1)p^2_{22}$$. For $$2\leq i\leq D-2$$ we define vectors $$w_{ii}^+=\sum |\varGamma_1(x)\cap\varGamma_1(y)\cap\varGamma_{i-1}(z)|\hat{z}$$, where the sum is over all vertices $$z\in\varGamma_i(x)\cap\varGamma_i(y)$$. We define $$W=\mathrm{span}\{w_{ij},w_{hh}^+ \mid 0\leq i,j\leq D,2\leq h\leq D-2\}$$. In [Discrete Math. 225, No. 1–3, 193–216 (2000; Zbl 1001.05124)], M. MacLean defined what it means for $$\varGamma$$ to be taut. Assume $$D$$ is odd. We show $$\varGamma$$ is taut if and only if $$\varDelta\neq 0$$ and the subspace $$W$$ is invariant under multiplication by the adjacency matrix.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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