A remark on the intersection arrays of distance regular graphs and the distance regular graphs of diameter \(d=3i -1\) with \(b_ i =1\) and \(k>2\) (Q1182424)

It is shown that in a distance regular graph [see the book of \textit{A. E. Brouwer}, \textit{A. M. Cohen} and \textit{A. Neumaier}, Distance Regular Graphs (Springer 1989; Zbl 0747.05073)] with intersection array \((b_ 0,b_ 1,\ldots,b_{d-1}, c_ 1,\ldots,c_ d)\), diameter \(d\geq 2i\), \(i\geq 2\), such that \((c_{i-1},a_{i-1},b_{i-1})=(c_ 1,a_ 1,b_ 1)\), \((c_{2i-1},a_{2i-1},b_{2i-1})=(c_ i,a_ i,b_ i)\), \(a_ 1=0\), \(c_ i=1\), \(a_ i>0\) and \(a_{2i}<a_ i\) the valency \(k\) must satisfy \(k\leq 1+2b_ i\). The graphs with \(d=3i-1\), \(b_ i=1\) and \(k>2\) are determined.