an:01319552
Zbl 0927.05085
Pascasio, Arlene A.
Tight graphs and their primitive idempotents
EN
J. Algebr. Comb. 10, No. 1, 47-59 (1999).
00057992
1999
j
05E30
tight graph; primitive idempotent; Krein parameter; distance regular graph; eigenvalues
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 \(k\theta_{d-1}= \theta_1\theta_d\)).
A.A.Makhnev (Ekaterinburg)