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Taut distance-regular graphs of even diameter. (English) Zbl 1047.05044
Let $$\Gamma$$ be a distance-regular graph with diameter $$D\geq 4$$ and valency $$k\geq 3$$. Let $$k=\theta_0>\cdots >\theta_D$$ denote the distinct eigenvalues of $$\Gamma$$, and for $$0\leq i\leq D$$, let $$E_i$$ denote the primitive idempotents of $$\Gamma$$ associated with $$\theta_i$$. A primitive idempotent $$F$$ of $$\Gamma$$ is called trivial, if $$F=E_0$$ or $$\Gamma$$ is bipartite and $$F=E_D$$. Let $$E,F$$ denote primitive idempotents of $$\Gamma$$. The pair $$E,F$$ is called taut whenever $$E,F$$ are nontrivial and the entry-wise product $$E\circ F$$ is a linear combination of two distinct primitive idempotents of $$\Gamma$$. We define $$\Gamma$$ to be taut, if $$\Gamma$$ is not 2-homogeneous and $$\Gamma$$ has the taut pair $$E,F$$ of primitive idempotents. Theorem. Let $$\Gamma$$ be a bipartite distance-regular graph with even diameter $$D\geq 4$$, valency $$k\geq 3$$, and eigenvalues $$\theta_0>\dots >\theta_D$$. Let $$\theta$$ denote an eigenvalue of $$\Gamma$$ other than $$\theta_0,\theta_D$$, and let $$\sigma_0,\dots ,\sigma_D$$ denote the cosine sequence associated with $$\theta$$. Then the following are equivalent: (i) $$\Gamma$$ is taut or 2-homogeneous, and $$\theta\in \{\theta_1,\theta_{D-1}\}$$. (ii) There exists a complex number $$\lambda$$ such that $$\sigma_{i-1}-\lambda \sigma_i+\sigma_{i+1}=0$$ for $$i$$ odd, $$1\leq i\leq D-1$$. (iii) There exists a complex number $$\lambda$$ such that $$\sigma_{i-1}-\lambda \sigma_i+\sigma_{i+1}=0$$ for $$i=1,3$$.

##### MSC:
 5e+30 Association schemes, strongly regular graphs
##### Keywords:
distance-regular graph; bipartite graph; tight graph; taut graph
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##### References:
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