A diameter 2 arc-transitive graph whose complement is not arc-transitive?

Question

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.

Answer 1

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 $v_2$.