Distance degrees of vertex-transitive graphs (Q917573)

Consider a connected distance degree regular graph (so that \(| \{v: d(u,v)=i\}| =k_ i,\) for all u) with \(k_ 1=3n\). It is known that \(k_ r\geq 2n+1,\) for \(2\leq r<diam G.\) It is shown here that if such a graph is vertex-transitive and \(k_ r=2n+1,\) for some r, then \(n=1\) and \(k_{r+1}=1\).