Vertex-imprimitive symmetric graphs with exactly one edge between any two distinct blocks

A graph Γ is called G-symmetric if it admits G as a group of automorphisms acting transitively on the set of ordered pairs of adjacent vertices. We give a classification of G-symmetric graphs Γ with V(Γ) admitting a nontrivial G-invariant partition B such that there is exactly one edge of Γ between any two distinct blocks of B. This is achieved by giving a classification of (G,2)-point-transitive and G-block-transitive designs D together with G-orbits Ω on the flag set of D such that Gσ,L is transitive on L∖{σ} and L∩N={σ} for distinct (σ,L),(σ,N)∈Ω, where Gσ,L is the setwise stabilizer of L in the stabilizer Gσ of σ in G. Along the way we determine all imprimitive blocks of Gσ on V∖{σ} for every 2-transitive group G on a set V, where σ∈V.