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Taut distance-regular graphs of odd diameter. (English) Zbl 1014.05072
A distance-regular graph $$\Gamma$$ with diameter $$D\geq 4$$, valency $$k\geq 3$$ and eigenvalues $$\theta_0>\cdots >\theta_D$$ is considered. Let $$M$$ denote the Bose-Mesner algebra of $$\Gamma$$. For $$0\leq i\leq D$$, let $$E_i$$ denote the primitive idempotents of $$M$$ associated with $$\theta_i$$ ($$E_0$$ and $$E_d$$ are called trivial idempotents). A pair $$E,F$$ of nontrivial primitive idempotents of $$M$$ is taut whenever the entry-wise product $$E\circ F$$ is a linear combination of two primitive idempotents of $$M$$. The graph $$\Gamma$$ is taut whenever $$\Gamma$$ has at least one taut pair of primitive idempotents, but $$\Gamma$$ is not 2-homogeneous in the sense of Nomura-Curtin.
In this paper the taut graphs of odd diameter $$D$$ are investigated. Let $$E,F$$ be nontrivial primitive idempotents of $$M$$ and let $$\sigma_0,\dots ,\sigma_d$$ and $$\rho_0,\dots ,\rho_d$$ denote the cosine sequences of $$E$$ and $$F$$, respectively. The pair $$E,F$$ is taut if and only if there exist real scalars $$\alpha$$ and $$\beta$$ such that $\sigma_{i+1}\rho_{i+1}-\sigma_{i-1}\rho_{i-1}= \alpha\sigma_i(\rho_{i+1}-\rho_{i-1})+\beta\rho_i(\sigma_{i+1}-\sigma_{i-1})\quad (1\leq i\leq D-1).$ If $$D$$ is odd and the pair $$E,F$$ is taut, then all intersection numbers of $$\Gamma$$ are determined in terms of $$\sigma_1,\rho_1,\alpha,\beta$$ (Theorem 5.8). As $$E_1,E_d$$ is a taut pair in any taut graph for $$d=(D-1)/2$$, all intersection numbers of taut graphs $$\Gamma$$ are determined in terms of $$k,\mu,\theta_1,\theta_d$$ (Corollary 5.9). If $$\Gamma$$ is taut and $$D$$ is odd, then $$\Gamma$$ is an antipodal 2-cover (Theorem 6.4).

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