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Tight graphs and their primitive idempotents. (English) Zbl 0927.05085
Let $$\Gamma$$ be a distance-regular graph with diameter $$d\geq 3$$ and eigenvalues $$\theta_0>\theta_1>\dots>\theta_d$$. Then $\Biggl( \theta_1+\frac{k}{a_1+1}\Biggr) \Biggl(\theta_d+\frac{k}{a_1+1}\Biggr)\geq \frac{-ka_1b_1}{(a_1+1)^2}.$ $$\Gamma$$ is said to be tight whenever $$\Gamma$$ is not bipartite and equality holds above. Suppose $$E$$ and $$F$$ are nontrivial primitive idempotents of $$\Gamma$$ and the entry-wise product $$E\circ F$$ is a scalar multiple of a primitive idempotent $$H$$ of $$\Gamma$$. Then $$\Gamma$$ is either bipartite and at least one of $$E,\;F$$ is equal to $$E_d$$ or tight and $$\{E,F\}=\{E_1,E_d\}$$ (in last case the eigenvalue associated with $$H$$ is $$\theta_{d-1}$$ and $$k\theta_{d-1}= \theta_1\theta_d$$).

##### MSC:
 5e+30 Association schemes, strongly regular graphs
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##### References:
 [1] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin-Cummings Lecture Note Ser. 58, Benjamin-Cummings, Menlo Park, CA. 1984. [2] A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer, New York, 1989. · Zbl 0747.05073 [3] C.D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993. [4] A. Jurišić, J. Koolen, and P. Terwilliger, “Tight distance-regular graphs,” submitted. · Zbl 0959.05121
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