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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 \(k\theta_{d-1}= \theta_1\theta_d\)).

05E30 Association schemes, strongly regular graphs
Full Text: DOI
[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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