On distance-regular graphs with \(k_ i=k_ j\) (Q1328390)

The following theorem is proved, which gives a affirmative solution to a conjecture in the book ``Distance-regular graphs'' written by \textit{A. E. Brouwer}, \textit{A. M. Cohen} and \textit{A. Neumaier} (Springer, 1989; Zbl 0747.05073). Theorem. Let \(\Gamma\) be a distance-regular graph with diameter \(d\). Suppose \(k_ e=k_ f\) with \(e<f\) and \(e+f \leq d\). Then one of the following holds: (i) \(\Gamma\) is an antipodal 2-cover, i.e., \(k_ d=1\); or (ii) \(k_ e = k_{e+1} = \cdots = k_ f\). Moreover, if \(k_ f \neq k_{f+1}\), then \(\Gamma_ d(u)\) is a clique for any vertex \(u\) in \(\Gamma\).