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

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.

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 $$v_2$$.