A graph $G=(V,E)$ is *arc-transitive* if its symmetry group acts transitively on ordered pairs of adjacent vertices.

In general, the complement of an arc-transitive graph is not arc-transitive. But I have a hard time finding an example of such a graph if I assume $\mathrm{diam}(G)=\max_{v,w\in V} \mathrm{dist}(v,w)=2$. All my examples of diameter 2 arc-transitive graphs have arc-transitive complements: e.g.

- the 4-cycle and 5-cycle,
- the Petersen-graph,
- the Hoffman-Singleton graph,
- the Payley graphs,
- ...

I suppose an equivalent question would be: find a diameter 2 arc-transitive graph that is *not* distance-transitive.