A graph $G=(V,E)$ is [*arc-transitive*](https://en.wikipedia.org/wiki/Symmetric_graph) 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](https://en.wikipedia.org/wiki/Distance-transitive_graph).



A diameter 2 arc-transitive graph whose complement is not arc-transitive?
Complements don't even have to be edge transitive. Perhaps the simplest example is the wreath graph $W\_5$ (which is obtained by applying the construction below to $C\_5$).


Call two vertices "twins", if they have the same neighbourhood in $G$. Since the twin-relation is preserved under automorphisms, it suffices to construct an arc-transitive graph $G$ of diameter $2$ which has non-adjacent twins as well as non-adjacent non-twins.


Now let $G$ be an arc-transitive graph $G$ of diameter $2$ and assume that there are non-adjacent non-twins $v\_1$ and $v\_2$ in $G$. Take the lexicographic product of $G$ with an empty graph on $2$ vertices. In other words, we add a twin $v'$ to every vertex $v$ of $G$, where $v'$ is connected to all neighbours of $v$ and their respective twins. The resulting graph is still arc transitive, has diameter $2$, and contains non-adjacent twins $v$ and $v'$ as well as non-adjacent non-twins $v\_1$ and $v\_2$.


