Imprimitive symmetric graphs with cyclic blocks

Let Γ be a graph admitting an arc-transitive subgroup G of automorphisms that leaves invariant a vertex partition B with parts of size v ≥ 3 . In this paper we study such graphs where: for B , C ∈ B connected by some edge of Γ , exactly two vertices of B lie on no edge with a vertex of C ; and as C runs over all parts of B connected to B these vertex pairs (ignoring multiplicities) form a cycle. We prove that this occurs if and only if v = 3 or 4, and moreover we give three geometric or group theoretic constructions of infinite families of such graphs.