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

Abstract

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 $\mathcal{B}$ such that there is exactly one edge of $Γ$ between any two distinct blocks of $\mathcal{B}$. This is achieved by giving a classification of $(G, 2)$-point-transitive and $G$-block-transitive designs $\mathcal{D}$ together with $G$-orbits $Ω$ on the flag set of $\mathcal{D}$ such that $G_{σ, L}$ is transitive on $L \setminus \{σ\}$ and $L \cap N = \{σ\}$ for distinct $(σ, L), (σ, N) \in Ω$, where $G_{σ, L}$ is the setwise stabilizer of $L$ in the stabilizer $G_σ$ of $σ$ in $G$. Along the way we determine all imprimitive blocks of $G_σ$ on $V \setminus \{σ\}$ for every $2$-transitive group $G$ on a set $V$, where $σ\in V$.