Distance-Uniform Graphs with Large Diameter

An $ε$-distance-uniform graph is one in which from every vertex, all but an $ε$-fraction of the remaining vertices are at some fixed distance $d$, called the critical distance. We consider the maximum possible value of $d$ in an $ε$-distance-uniform graph with $n$ vertices. We show that for $\frac1n \le ε\le \frac1{\log n}$, there exist $ε$-distance-uniform graphs with critical distance $2^{Ω(\frac{\log n}{\log ε^{-1}})}$, disproving a conjecture of Alon et al. that $d$ can be at most logarithmic in $n$. We also show that our construction is best possible, in the sense that an upper bound on $d$ of the form $2^{O(\frac{\log n}{\log ε^{-1}})}$ holds for all $ε$ and $n$.